Explanation:
Okay, so I need to simplify the expression (1 + √2)/(2 + √3). The problem says that the simplified form can be written as A(1 + √B) - (√C + √D), where A, B, C, D are positive integers, and then find A + B + C + D. Alright, let me think about how to approach this.

First, I remember that when you have a fraction with radicals in the denominator, you can rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of a binomial like (a + b) is (a - b), right? So in this case, the denominator is 2 + √3, so its conjugate would be 2 - √3. Let me try that.

So, multiplying numerator and denominator by 2 - √3:

[(1 + √2)(2 - √3)] / [(2 + √3)(2 - √3)]

Let me compute the denominator first because that might be simpler. The denominator is a difference of squares: (2)^2 - (√3)^2 = 4 - 3 = 1. Wait, that's nice! The denominator simplifies to 1? That would make the whole expression just equal to the numerator. Wait, let me check that again.

Yes, (2 + √3)(2 - √3) = 2^2 - (√3)^2 = 4 - 3 = 1. So the denominator is 1, so the entire expression simplifies to just the numerator: (1 + √2)(2 - √3). Hmm, so maybe I need to expand that numerator?

Let me do that. Let's multiply (1 + √2)(2 - √3). Using the distributive property (FOIL):

First: 1 * 2 = 2

Outer: 1 * (-√3) = -√3

Inner: √2 * 2 = 2√2

Last: √2 * (-√3) = -√6

So combining all those terms: 2 - √3 + 2√2 - √6. So the entire expression simplifies to 2 - √3 + 2√2 - √6. Wait, but the problem says the simplified form can be written as A(1 + √B) - (√C + √D). Let me see if I can rearrange the terms I have here to match that form.

So, the expression is 2 - √3 + 2√2 - √6. Let me group the terms. Let me see:

First, maybe group the constants and the radicals. But there's a 2, then -√3, then +2√2, then -√6. Let me think. The target form is A(1 + √B) - (√C + √D). Let me think about how to express this.

Alternatively, maybe I made a mistake in the simplification? Let me check again. Wait, the original expression was (1 + √2)/(2 + √3). After rationalizing, we multiplied numerator and denominator by 2 - √3, denominator becomes 1, numerator is (1 + √2)(2 - √3). Let me check that multiplication again.

Yes, expanding (1 + √2)(2 - √3):

1*2 = 2

1*(-√3) = -√3

√2*2 = 2√2

√2*(-√3) = -√6

So total is 2 - √3 + 2√2 - √6. That seems correct.

So, the simplified expression is 2 - √3 + 2√2 - √6. Now, how can I write this in the form A(1 + √B) - (√C + √D)?

Let me think. Let me try to group terms. Let me see:

Looking at the expression: 2 - √3 + 2√2 - √6. Let me see if I can factor something out. For example, maybe group the 2 and 2√2 together? Let me try:

2 + 2√2 - √3 - √6. Then, factor 2 from the first two terms: 2(1 + √2) - (√3 + √6). Wait, that seems promising. Let me check:

Yes, 2(1 + √2) - (√3 + √6). But the problem says the form is A(1 + √B) - (√C + √D). So here, A would be 2, B would be 2, and then √C + √D would be √3 + √6. But then C and D would be 3 and 6. However, the problem says that A, B, C, D are positive integers. So that would make A=2, B=2, C=3, D=6. Then A+B+C+D would be 2+2+3+6=13. But wait, let me check if that's correct.

Wait, but let me check if the original expression is equal to 2(1 + √2) - (√3 + √6). Let me compute that:

2(1 + √2) = 2 + 2√2

Then subtract (√3 + √6): 2 + 2√2 - √3 - √6. Which is exactly the same as the simplified expression. So that seems correct. Therefore, A=2, B=2, C=3, D=6. Then A+B+C+D=2+2+3+6=13. Is that the answer? Wait, but let me check again if there's another way to write it, maybe with different A, B, C, D? Because sometimes there might be multiple forms, but the problem says "the form A(1 + √B) - (√C + √D)", so maybe this is the only way? Let me check.

Alternatively, maybe there's a different grouping? Let me think. Suppose I tried to write the expression as something else. For example, maybe combining the -√3 and -√6? But that would be -√3 - √6, which is the same as -(√3 + √6). So that's the same as before. So the expression is 2 + 2√2 - √3 - √6, which is 2(1 + √2) - (√3 + √6). So that seems to fit the required form. Therefore, A=2, B=2, C=3, D=6. Then A+B+C+D=2+2+3+6=13. But let me check again if that's correct.

Wait, but let me check if the original problem had a different answer. Wait, maybe I made a mistake in rationalizing? Let me check again. Original expression: (1 + √2)/(2 + √3). Multiply numerator and denominator by 2 - √3:

Numerator: (1 + √2)(2 - √3) = 2 - √3 + 2√2 - √6. Denominator: 1. So yes, that's correct. So the simplified expression is 2 - √3 + 2√2 - √6. Which is equal to 2(1 + √2) - (√3 + √6). Therefore, A=2, B=2, C=3, D=6. So the sum is 2+2+3+6=13. But let me check if there's another way to write this expression. For example, maybe combining √6 with something else?

Wait, √6 can be written as √2*√3. But I don't know if that helps. Let me think. Suppose I tried to write the expression as A(1 + √B) - (√C + √D). Let me think if there's another way. For example, maybe grouping terms differently. Let me see:

Original expression after simplifying: 2 - √3 + 2√2 - √6. Let me think of 2 as 1 + 1. Then maybe 1 + 1 + 2√2 - √3 - √6. But that doesn't seem helpful. Alternatively, maybe factor something else. Let me think. Suppose I want to write it as A(1 + √B) - (√C + √D). Let me think of the terms:

We have 2 + 2√2 - √3 - √6. Let me think of 2 + 2√2 as 2(1 + √2). Then the rest is -√3 - √6. So that's the same as before. So that seems to be the only way. Therefore, A=2, B=2, C=3, D=6. Therefore, the answer is 2+2+3+6=13. But let me check again if that's correct. Wait, but maybe there's a different way to rationalize? Let me think. Wait, maybe I should have rationalized differently? Let me check.

Alternatively, suppose I didn't rationalize the denominator? But that would complicate things. The standard method is to rationalize, so I think that's the right approach. Alternatively, maybe there's a mistake in the problem statement? Wait, the problem says "Your solution can be converted to the form A(1 + √B) - (√C + √D)". So maybe there's another way? Let me think.

Alternatively, suppose I tried to write the expression as something else. Let me think. Let me check if 2(1 + √2) - (√3 + √6) can be simplified further or written in another way. For example, maybe √6 can be expressed as √2*√3, but I don't know if that helps. Let me think. Suppose I want to write the expression as A(1 + √B) - (√C + √D). If I have 2(1 + √2) - (√3 + √6), then that's already in the required form. So unless there's another way to factor or combine terms, this seems to be the answer.

But let me check if the problem expects different values. For example, maybe A is different? Let me think. Suppose I tried to write the expression as something else. Let me think. Let me check if there's another way to group terms. For example, suppose I take 2 - √3 and 2√2 - √6. Let me see:

2 - √3 + 2√2 - √6. Let me factor something from 2√2 - √6. Let me see, 2√2 - √6 = √2(2 - √3). Wait, that's interesting. Let me check:

√2*(2 - √3) = 2√2 - √2*√3 = 2√2 - √6. Yes! So that's correct. Therefore, the entire expression can be written as (2 - √3) + √2*(2 - √3) = (2 - √3)(1 + √2). Wait, that's interesting. So that would be (2 - √3)(1 + √2). But that's the original numerator before simplifying. Wait, but we had already simplified it to 2 - √3 + 2√2 - √6. But if we factor (2 - √3)(1 + √2), that's equal to the original expression. Wait, but that's the same as before. So that's not helpful for the required form. However, if I write it as (2 - √3)(1 + √2), but the problem wants it in the form A(1 + √B) - (√C + √D). So maybe that's not helpful. Let me think.

Alternatively, if I take (2 - √3)(1 + √2) and expand it, which we already did. So that gives us the same expression. So maybe that's not helpful. So perhaps my initial thought is correct. Let me check again.

Original simplified expression: 2 - √3 + 2√2 - √6. Which is equal to 2(1 + √2) - (√3 + √6). Therefore, A=2, B=2, C=3, D=6. Therefore, A+B+C+D=2+2+3+6=13. But let me check if that's the answer. Wait, but maybe there's a different way to write it? Let me think. For example, if I can write √6 as √something else? But √6 is already simplified. Similarly, √3 is simplified. So I don't think there's a way to combine them further. Therefore, I think that this is the correct answer.

But let me check with another approach. Let me suppose that the answer is supposed to be in the form A(1 + √B) - (√C + √D). Let me assume that and try to find A, B, C, D. Let me denote the expression as:

A(1 + √B) - (√C + √D) = A + A√B - √C - √D.

Compare this with the simplified expression: 2 - √3 + 2√2 - √6. So, let me equate the terms:

The constant terms: A = 2.

The terms with √B: A√B = 2√2. Since A=2, then 2√B = 2√2 ⇒ √B = √2 ⇒ B=2.

Then, the terms with radicals: -√C - √D = -√3 - √6. Therefore, √C + √D = √3 + √6. Therefore, C and D are 3 and 6. Since they are positive integers, and order doesn't matter (since addition is commutative), so C=3 and D=6. Therefore, A=2, B=2, C=3, D=6. Therefore, the sum is 2+2+3+6=13. So that seems consistent.

Therefore, I think the answer is 13. But let me check once more to make sure I didn't make any mistakes. Let me compute the original expression numerically and the simplified expression numerically to check if they are equal.

Original expression: (1 + √2)/(2 + √3). Let me compute approximate values.

√2 ≈ 1.4142, √3 ≈ 1.732.

So numerator: 1 + 1.4142 ≈ 2.4142.

Denominator: 2 + 1.732 ≈ 3.732.

So original expression ≈ 2.4142 / 3.732 ≈ Let me compute that. Let me divide 2.4142 by 3.732. Let me approximate:

Well, 3.732 * 0.6 = 2.2392. 2.4142 - 2.2392 = 0.175. So 0.6 + 0.175 / 3.732 ≈ 0.6 + 0.0469 ≈ 0.6469. So approximately 0.6469.

Now, let me compute the simplified expression: 2 - √3 + 2√2 - √6.

Compute each term:

2 = 2.

-√3 ≈ -1.732.

2√2 ≈ 2 * 1.4142 ≈ 2.8284.

-√6 ≈ -2.4495.

Adding them up: 2 - 1.732 + 2.8284 - 2.4495 ≈ Let's compute step by step:

Start with 2 - 1.732 = 0.268.

Then add 2.8284: 0.268 + 2.8284 ≈ 3.0964.

Then subtract 2.4495: 3.0964 - 2.4495 ≈ 0.6469. Which matches the original expression's approximate value. Therefore, the simplified expression is correct. Therefore, my answer of A=2, B=2, C=3, D=6 is correct, and the sum is 13.

But just to be thorough, let me check if there's another possible combination. For example, suppose someone tried to write the expression as A(1 + √B) - (√C + √D) with different values. Let me think. For example, could there be another way to group terms?

Suppose I tried to write the expression as something like (something) - (something else). Let me think. For example, if I take 2 - √3 - √6 + 2√2. Maybe factor something else. For example, 2 - √3 - √6 + 2√2. Let me see if I can factor √2 from some terms. Let me see:

2√2 is already there. Then, -√6 is -√2*√3. So maybe:

2 - √3 + √2(2 - √3). Which is similar to before. Then, factor (2 - √3):

(2 - √3)(1 + √2). Which is the same as before. So again, same result. Therefore, no other way. Therefore, I think my answer is correct.

Therefore, the answer is 13. So A + B + C + D = 2 + 2 + 3 + 6 = 13.

**Final Answer**
\boxed{13}

To simplify the expression
$$
\frac{1 + \sqrt{2}}{2 + \sqrt{3}},
$$
we begin by **rationalizing the denominator**. The denominator is $2 + \sqrt{3}$, and its conjugate is $2 - \sqrt{3}$. Multiplying both numerator and denominator by this conjugate:

$$
\frac{1 + \sqrt{2}}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{(1 + \sqrt{2})(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})}.
$$

---

### Step 1: Simplify the Denominator

$$
(2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1.
$$

So the denominator becomes 1, and the expression simplifies to just the numerator:

$$
(1 + \sqrt{2})(2 - \sqrt{3}).
$$

---

### Step 2: Expand the Numerator

Use the distributive property (FOIL):

$$
(1 + \sqrt{2})(2 - \sqrt{3}) = 1 \cdot 2 + 1 \cdot (-\sqrt{3}) + \sqrt{2} \cdot 2 + \sqrt{2} \cdot (-\sqrt{3}) = 2 - \sqrt{3} + 2\sqrt{2} - \sqrt{6}.
$$

So the simplified expression is:

$$
2 - \sqrt{3} + 2\sqrt{2} - \sqrt{6}.
$$

---

### Step 3: Match the Form $A(1 + \sqrt{B}) - (\sqrt{C} + \sqrt{D})$

We want to express the simplified expression in the form:

$$
A(1 + \sqrt{B}) - (\sqrt{C} + \sqrt{D}).
$$

Group the terms:

$$
2 + 2\sqrt{2} - \sqrt{3} - \sqrt{6} = 2(1 + \sqrt{2}) - (\sqrt{3} + \sqrt{6}).
$$

This matches the desired form with:

- $A = 2$
- $B = 2$
- $C = 3$
- $D = 6$

---

### Step 4: Compute the Final Answer

$$
A + B + C + D = 2 + 2 + 3 + 6 = \boxed{13}
$$

[Assistant]
Explanation:
Okay, so I need to simplify the expression (1 + √2)/(2 + √3). The problem says that the simplified form can be written as A(1 + √B) - (√C + √D), where A, B, C, D are positive integers, and then find A + B + C + D. Alright, let me think about how to approach this.

First, I remember that when you have a fraction with radicals in the denominator, you can rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of a binomial like (a + b) is (a - b), right? So in this case, the denominator is 2 + √3, so its conjugate would be 2 - √3. Let me try that.

So, multiplying numerator and denominator by 2 - √3:

[(1 + √2)(2 - √3)] / [(2 + √3)(2 - √3)]

Let me compute the denominator first because that might be simpler. The denominator is a difference of squares: (2)^2 - (√3)^2 = 4 - 3 = 1. Wait, that's nice! The denominator simplifies to 1? That would make the whole expression just equal to the numerator. Wait, let me check that again.

Yes, (2 + √3)(2 - √3) = 2^2 - (√3)^2 = 4 - 3 = 1. So the denominator is 1, so the entire expression simplifies to just the numerator: (1 + √2)(2 - √3). Hmm, so maybe I need to expand that numerator?

Let me do that. Let's multiply (1 + √2)(2 - √3). Using the distributive property (FOIL):

First: 1 * 2 = 2

Outer: 1 * (-√3) = -√3

Inner: √2 * 2 = 2√2

Last: √2 * (-√3) = -√6

So combining all those terms: 2 - √3 + 2√2 - √6. So the entire expression simplifies to 2 - √3 + 2√2 - √6. Wait, but the problem says the simplified form can be written as A(1 + √B) - (√C + √D). Let me see if I can rearrange the terms I have here to match that form.

So, the expression is 2 - √3 + 2√2 - √6. Let me group the terms. Let me see:

First, maybe group the constants and the radicals. But there's a 2, then -√3, then +2√2, then -√6. Let me think. The target form is A(1 + √B) - (√C + √D). Let me think about how to express this.

Alternatively, maybe I made a mistake in the simplification? Let me check again. Wait, the original expression was (1 + √2)/(2 + √3). After rationalizing, we multiplied numerator and denominator by 2 - √3, denominator becomes 1, numerator is (1 + √2)(2 - √3). Let me check that multiplication again.

Yes, expanding (1 + √2)(2 - √3):

1*2 = 2

1*(-√3) = -√3

√2*2 = 2√2

√2*(-√3) = -√6

So total is 2 - √3 + 2√2 - √6. That seems correct.

So, the simplified expression is 2 - √3 + 2√2 - √6. Now, how can I write this in the form A(1 + √B) - (√C + √D)?

Let me think. Let me try to group terms. Let me see:

Looking at the expression: 2 - √3 + 2√2 - √6. Let me see if I can factor something out. For example, maybe group the 2 and 2√2 together? Let me try:

2 + 2√2 - √3 - √6. Then, factor 2 from the first two terms: 2(1 + √2) - (√3 + √6). Wait, that seems promising. Let me check:

Yes, 2(1 + √2) - (√3 + √6). But the problem says the form is A(1 + √B) - (√C + √D). So here, A would be 2, B would be 2, and then √C + √D would be √3 + √6. But then C and D would be 3 and 6. However, the problem says that A, B, C, D are positive integers. So that would make A=2, B=2, C=3, D=6. Then A+B+C+D would be 2+2+3+6=13. But wait, let me check if that's correct.

Wait, but let me check if the original expression is equal to 2(1 + √2) - (√3 + √6). Let me compute that:

2(1 + √2) = 2 + 2√2

Then subtract (√3 + √6): 2 + 2√2 - √3 - √6. Which is exactly the same as the simplified expression. So that seems correct. Therefore, A=2, B=2, C=3, D=6. Then A+B+C+D=2+2+3+6=13. Is that the answer? Wait, but let me check again if there's another way to write it, maybe with different A, B, C, D? Because sometimes there might be multiple forms, but the problem says "the form A(1 + √B) - (√C + √D)", so maybe this is the only way? Let me check.

Alternatively, maybe there's a different grouping? Let me think. Suppose I tried to write the expression as something else. For example, maybe combining the -√3 and -√6? But that would be -√3 - √6, which is the same as -(√3 + √6). So that's the same as before. So the expression is 2 + 2√2 - √3 - √6, which is 2(1 + √2) - (√3 + √6). So that seems to fit the required form. Therefore, A=2, B=2, C=3, D=6. Then A+B+C+D=2+2+3+6=13. But let me check again if that's correct.

Wait, but let me check if the original problem had a different answer. Wait, maybe I made a mistake in rationalizing? Let me check again. Original expression: (1 + √2)/(2 + √3). Multiply numerator and denominator by 2 - √3:

Numerator: (1 + √2)(2 - √3) = 2 - √3 + 2√2 - √6. Denominator: 1. So yes, that's correct. So the simplified expression is 2 - √3 + 2√2 - √6. Which is equal to 2(1 + √2) - (√3 + √6). Therefore, A=2, B=2, C=3, D=6. So the sum is 2+2+3+6=13. But let me check if there's another way to write this expression. For example, maybe combining √6 with something else?

Wait, √6 can be written as √2*√3. But I don't know if that helps. Let me think. Suppose I tried to write the expression as A(1 + √B) - (√C + √D). Let me think if there's another way. For example, maybe grouping terms differently. Let me see:

Original expression after simplifying: 2 - √3 + 2√2 - √6. Let me think of 2 as 1 + 1. Then maybe 1 + 1 + 2√2 - √3 - √6. But that doesn't seem helpful. Alternatively, maybe factor something else. Let me think. Suppose I want to write it as A(1 + √B) - (√C + √D). Let me think of the terms:

We have 2 + 2√2 - √3 - √6. Let me think of 2 + 2√2 as 2(1 + √2). Then the rest is -√3 - √6. So that's the same as before. So that seems to be the only way. Therefore, A=2, B=2, C=3, D=6. Therefore, the answer is 2+2+3+6=13. But let me check again if that's correct. Wait, but maybe there's a different way to rationalize? Let me think. Wait, maybe I should have rationalized differently? Let me check.

Alternatively, suppose I didn't rationalize the denominator? But that would complicate things. The standard method is to rationalize, so I think that's the right approach. Alternatively, maybe there's a mistake in the problem statement? Wait, the problem says "Your solution can be converted to the form A(1 + √B) - (√C + √D)". So maybe there's another way? Let me think.

Alternatively, suppose I tried to write the expression as something else. Let me think. Let me check if 2(1 + √2) - (√3 + √6) can be simplified further or written in another way. For example, maybe √6 can be expressed as √2*√3, but I don't know if that helps. Let me think. Suppose I want to write the expression as A(1 + √B) - (√C + √D). If I have 2(1 + √2) - (√3 + √6), then that's already in the required form. So unless there's another way to factor or combine terms, this seems to be the answer.

But let me check if the problem expects different values. For example, maybe A is different? Let me think. Suppose I tried to write the expression as something else. Let me think. Let me check if there's another way to group terms. For example, suppose I take 2 - √3 and 2√2 - √6. Let me see:

2 - √3 + 2√2 - √6. Let me factor something from 2√2 - √6. Let me see, 2√2 - √6 = √2(2 - √3). Wait, that's interesting. Let me check:

√2*(2 - √3) = 2√2 - √2*√3 = 2√2 - √6. Yes! So that's correct. Therefore, the entire expression can be written as (2 - √3) + √2*(2 - √3) = (2 - √3)(1 + √2). Wait, that's interesting. So that would be (2 - √3)(1 + √2). But that's the original numerator before simplifying. Wait, but we had already simplified it to 2 - √3 + 2√2 - √6. But if we factor (2 - √3)(1 + √2), that's equal to the original expression. Wait, but that's the same as before. So that's not helpful for the required form. However, if I write it as (2 - √3)(1 + √2), but the problem wants it in the form A(1 + √B) - (√C + √D). So maybe that's not helpful. Let me think.

Alternatively, if I take (2 - √3)(1 + √2) and expand it, which we already did. So that gives us the same expression. So maybe that's not helpful. So perhaps my initial thought is correct. Let me check again.

Original simplified expression: 2 - √3 + 2√2 - √6. Which is equal to 2(1 + √2) - (√3 + √6). Therefore, A=2, B=2, C=3, D=6. Therefore, A+B+C+D=2+2+3+6=13. But let me check if that's the answer. Wait, but maybe there's a different way to write it? Let me think. For example, if I can write √6 as √something else? But √6 is already simplified. Similarly, √3 is simplified. So I don't think there's a way to combine them further. Therefore, I think that this is the correct answer.

But let me check with another approach. Let me suppose that the answer is supposed to be in the form A(1 + √B) - (√C + √D). Let me assume that and try to find A, B, C, D. Let me denote the expression as:

A(1 + √B) - (√C + √D) = A + A√B - √C - √D.

Compare this with the simplified expression: 2 - √3 + 2√2 - √6. So, let me equate the terms:

The constant terms: A = 2.

The terms with √B: A√B = 2√2. Since A=2, then 2√B = 2√2 ⇒ √B = √2 ⇒ B=2.

Then, the terms with radicals: -√C - √D = -√3 - √6. Therefore, √C + √D = √3 + √6. Therefore, C and D are 3 and 6. Since they are positive integers, and order doesn't matter (since addition is commutative), so C=3 and D=6. Therefore, A=2, B=2, C=3, D=6. Therefore, the sum is 2+2+3+6=13. So that seems consistent.

Therefore, I think the answer is 13. But let me check once more to make sure I didn't make any mistakes. Let me compute the original expression numerically and the simplified expression numerically to check if they are equal.

Original expression: (1 + √2)/(2 + √3). Let me compute approximate values.

√2 ≈ 1.4142, √3 ≈ 1.732.

So numerator: 1 + 1.4142 ≈ 2.4142.

Denominator: 2 + 1.732 ≈ 3.732.

So original expression ≈ 2.4142 / 3.732 ≈ Let me compute that. Let me divide 2.4142 by 3.732. Let me approximate:

Well, 3.732 * 0.6 = 2.2392. 2.4142 - 2.2392 = 0.175. So 0.6 + 0.175 / 3.732 ≈ 0.6 + 0.0469 ≈ 0.6469. So approximately 0.6469.

Now, let me compute the simplified expression: 2 - √3 + 2√2 - √6.

Compute each term:

2 = 2.

-√3 ≈ -1.732.

2√2 ≈ 2 * 1.4142 ≈ 2.8284.

-√6 ≈ -2.4495.

Adding them up: 2 - 1.732 + 2.8284 - 2.4495 ≈ Let's compute step by step:

Start with 2 - 1.732 = 0.268.

Then add 2.8284: 0.268 + 2.8284 ≈ 3.0964.

Then subtract 2.4495: 3.0964 - 2.4495 ≈ 0.6469. Which matches the original expression's approximate value. Therefore, the simplified expression is correct. Therefore, my answer of A=2, B=2, C=3, D=6 is correct, and the sum is 13.

But just to be thorough, let me check if there's another possible combination. For example, suppose someone tried to write the expression as A(1 + √B) - (√C + √D) with different values. Let me think. For example, could there be another way to group terms?

Suppose I tried to write the expression as something like (something) - (something else). Let me think. For example, if I take 2 - √3 - √6 + 2√2. Maybe factor something else. For example, 2 - √3 - √6 + 2√2. Let me see if I can factor √2 from some terms. Let me see:

2√2 is already there. Then, -√6 is -√2*√3. So maybe:

2 - √3 + √2(2 - √3). Which is similar to before. Then, factor (2 - √3):

(2 - √3)(1 + √2). Which is the same as before. So again, same result. Therefore, no other way. Therefore, I think my answer is correct.

Therefore, the answer is 13. So A + B + C + D = 2 + 2 + 3 + 6 = 13.

**Final Answer**
\boxed{13}

To simplify the expression
$$
\frac{1 + \sqrt{2}}{2 + \sqrt{3}},
$$
we begin by **rationalizing the denominator**. The denominator is $2 + \sqrt{3}$, and its conjugate is $2 - \sqrt{3}$. Multiplying both numerator and denominator by this conjugate:

$$
\frac{1 + \sqrt{2}}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{(1 + \sqrt{2})(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})}.
$$

---

### Step 1: Simplify the Denominator

$$
(2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1.
$$

So the denominator becomes 1, and the expression simplifies to just the numerator:

$$
(1 + \sqrt{2})(2 - \sqrt{3}).
$$

---

### Step 2: Expand the Numerator

Use the distributive property (FOIL):

$$
(1 + \sqrt{2})(2 - \sqrt{3}) = 1 \cdot 2 + 1 \cdot (-\sqrt{3}) + \sqrt{2} \cdot 2 + \sqrt{2} \cdot (-\sqrt{3}) = 2 - \sqrt{3} + 2\sqrt{2} - \sqrt{6}.
$$

So the simplified expression is:

$$
2 - \sqrt{3} + 2\sqrt{2} - \sqrt{6}.
$$

---

### Step 3: Match the Form $A(1 + \sqrt{B}) - (\sqrt{C} + \sqrt{D})$

We want to express the simplified expression in the form:

$$
A(1 + \sqrt{B}) - (\sqrt{C} + \sqrt{D}).
$$

Group the terms:

$$
2 + 2\sqrt{2} - \sqrt{3} - \sqrt{6} = 2(1 + \sqrt{2}) - (\sqrt{3} + \sqrt{6}).
$$

This matches the desired form with:

- $A = 2$
- $B = 2$
- $C = 3$
- $D = 6$

---

### Step 4: Compute the Final Answer

$$
A + B + C + D = 2 + 2 + 3 + 6 = \boxed{13}
$$

Solution: Multiplying top and bottom by the conjugate, we have $\frac{1+\sqrt{2}}{2+\sqrt{3}} = \frac{(1+\sqrt{2})(2-\sqrt{3})}{(2+\sqrt{3})((2-\sqrt{3}))} = \frac{2-\sqrt{3}+2\sqrt{2}-\sqrt{6}}{4-3} = 2-\sqrt{3}+2\sqrt{2}-\sqrt{6}$. So, we obtain $A=2, B=2, C=3$ and $D=6$ ($C$ and $D$ are interchangeable). So $A+B+C+D = 2+2+3+6 = \boxed{13}$.


Explanation:
Okay, so I need to find the area of quadrilateral DBEF. Let me first try to visualize the diagram based on the Asymptote code provided. There's a coordinate system with points A at (0,6), B at (0,0), and C at (8,0). D is the midpoint of AB, so since AB goes from (0,6) to (0,0), the midpoint D should be at (0,3). Similarly, E is the midpoint of BC, which goes from (0,0) to (8,0), so midpoint E is at (4,0). Then there's a point F at (8/3, 2).

The diagram also shows some lines: E connected to A, A connected to C, C connected to D, and there are axes with labels x and y. The quadrilateral in question is DBEF. Let me note the coordinates of all the points involved:

- D is (0,3)
- B is (0,0)
- E is (4,0)
- F is (8/3, 2)

So quadrilateral DBEF has vertices at D(0,3), B(0,0), E(4,0), and F(8/3, 2). To find its area, I can use coordinate geometry methods. One common method is the shoelace formula, which allows calculating the area of a polygon when the coordinates of the vertices are known.

First, let me recall the shoelace formula. For a polygon with vertices (x1, y1), (x2, y2), ..., (xn, yn), the area is given by:

Area = |(x1y2 - x2y1 + x2y3 - x3y2 + ... + xn y1 - x1 yn)/2|

So I need to list the coordinates of the quadrilateral DBEF in order. Let me make sure the points are ordered either clockwise or counter-clockwise. Let me check the positions:

- D is at (0,3)
- B is at (0,0)
- E is at (4,0)
- F is at (8/3, 2)

If I connect D to B to E to F and back to D, that should form the quadrilateral. Let me confirm the order. Starting at D(0,3), going to B(0,0), then to E(4,0), then to F(8/3, 2), and back to D(0,3). That seems to make a quadrilateral. Let me visualize this: from D down to B, then right to E, then up to F, then back to D. That should be a four-sided figure.

Alternatively, maybe I should check if the order is correct. If I list them in order, they should be either clockwise or counter-clockwise without crossing. Let me think. Starting at D(0,3), moving to B(0,0) is straight down. Then to E(4,0) is moving right along the x-axis. Then to F(8/3, 2) is moving up and to the left? Wait, 8/3 is approximately 2.666, so from E(4,0) to F(8/3, 2) would be moving left to x=8/3 (which is about 2.666) and up to y=2. Then back to D(0,3). That seems okay. So the order D-B-E-F-D is correct.

Alternatively, maybe I should list them in a different order? Let me check if there's a possibility of crossing lines. If I connect D-B-E-F-D, the sides are DB, BE, EF, FD. Let me see:

- DB is from D(0,3) to B(0,0)
- BE is from B(0,0) to E(4,0)
- EF is from E(4,0) to F(8/3, 2)
- FD is from F(8/3, 2) back to D(0,3)

Yes, that seems to form a quadrilateral without crossing. So the shoelace formula should work here.

Alternatively, maybe I can divide the quadrilateral into triangles or other shapes whose areas I can compute and add up. But since I have coordinates, shoelace might be straightforward.

Let me proceed with the shoelace formula. Let me list the coordinates in order:

D(0,3), B(0,0), E(4,0), F(8/3, 2), and back to D(0,3).

So writing them down:

x: 0, 0, 4, 8/3, 0

y: 3, 0, 0, 2, 3

Now applying the shoelace formula:

Compute sum of x_i * y_{i+1}:

First term: x1*y2 = 0*0 = 0

Second term: x2*y3 = 0*0 = 0

Third term: x3*y4 = 4*2 = 8

Fourth term: x4*y5 = (8/3)*3 = 8

Sum of these terms: 0 + 0 + 8 + 8 = 16

Now compute sum of y_i * x_{i+1}:

First term: y1*x2 = 3*0 = 0

Second term: y2*x3 = 0*4 = 0

Third term: y3*x4 = 0*(8/3) = 0

Fourth term: y4*x5 = 2*0 = 0

Sum of these terms: 0 + 0 + 0 + 0 = 0

Wait, that can't be right. The area would be |16 - 0| / 2 = 8. But that seems too large? Let me check my calculations again. Maybe I messed up the order of the points or the multiplication.

Wait, let me redo the shoelace formula step by step. Maybe I made a mistake in the order or the terms.

The shoelace formula is:

Area = 1/2 |sum_{i=1 to n} (x_i y_{i+1} - x_{i+1} y_i)|

So perhaps I need to compute each term (x_i y_{i+1} - x_{i+1} y_i) and sum them up.

Let me list the coordinates again:

Point 1: D(0, 3)

Point 2: B(0, 0)

Point 3: E(4, 0)

Point 4: F(8/3, 2)

Point 5: D(0, 3) [closing the polygon]

So compute each term:

Term 1: x1 y2 - x2 y1 = 0*0 - 0*3 = 0 - 0 = 0

Term 2: x2 y3 - x3 y2 = 0*0 - 4*0 = 0 - 0 = 0

Term 3: x3 y4 - x4 y3 = 4*2 - (8/3)*0 = 8 - 0 = 8

Term 4: x4 y5 - x5 y4 = (8/3)*3 - 0*2 = 8 - 0 = 8

Sum of terms: 0 + 0 + 8 + 8 = 16

Then area is 1/2 * |16| = 8. So area is 8? But let me check if that's correct. Wait, the entire triangle ABC has area?

Let me compute the area of triangle ABC to see if 8 is reasonable. Coordinates A(0,6), B(0,0), C(8,0). The area is (base * height)/2 = (8 * 6)/2 = 24. So quadrilateral DBEF has area 8? That would be a third of the total area. Maybe? Let me check with another method.

Alternatively, maybe I made a mistake in the order of the points? Let me try a different order. Maybe the points are not ordered correctly for the shoelace formula. Let me try ordering them differently. For example, maybe D, F, E, B? Or some other order?

Wait, let me think. If I traverse the quadrilateral in a different order, maybe I get a different result? Let me try.

Alternatively, maybe I should plot the points mentally. D is at (0,3), B is at (0,0), E is at (4,0), F is at (8/3, 2). So quadrilateral DBEF. Let me imagine connecting D to B to E to F to D. So from D(0,3) down to B(0,0), then right to E(4,0), then up to F(8/3, 2), then back to D(0,3).

Alternatively, maybe I can divide the quadrilateral into two triangles: DBE and DEF? Or maybe another combination?

Wait, let me try coordinates again. Maybe I made a mistake in the shoelace formula. Let me check again.

Alternatively, perhaps I should use vectors or coordinate geometry to find the area.

Alternatively, maybe I can use coordinates to find equations of lines and find intersections? Wait, but the coordinates of F are given as (8/3, 2). So maybe F is the intersection of some lines?

Looking back at the Asymptote code, the diagram draws E--A--C--D. So E is connected to A, A is connected to C, and C is connected to D. Also, there are axes. So maybe F is the intersection of EA and CD?

Wait, let me check. If E is (4,0) and A is (0,6), then line EA can be found. Similarly, line CD connects C(8,0) to D(0,3). Let me find the intersection point F of EA and CD. Maybe that's how F is defined?

Yes, the Asymptote code draws E--A--C--D, so maybe F is the intersection of EA and CD? Let me verify that.

So let me find equations for lines EA and CD.

First, line EA: connects E(4,0) to A(0,6). The slope of EA is (6 - 0)/(0 - 4) = 6 / (-4) = -3/2. So equation is y - 0 = -3/2(x - 4), which simplifies to y = -3/2 x + 6.

Line CD connects C(8,0) to D(0,3). The slope is (3 - 0)/(0 - 8) = 3 / (-8) = -3/8. Equation: y - 0 = -3/8(x - 8), so y = -3/8 x + 3.

Find intersection point F of EA and CD. So set the two equations equal:

-3/2 x + 6 = -3/8 x + 3

Multiply both sides by 8 to eliminate denominators:

-12x + 48 = -3x + 24

Bring all terms to left:

-12x + 48 + 3x - 24 = 0 => -9x + 24 = 0 => -9x = -24 => x = 24/9 = 8/3. Then y = -3/2*(8/3) + 6 = -4 + 6 = 2. So yes, F is (8/3, 2). So that's correct. Therefore, F is the intersection of EA and CD.

Therefore, quadrilateral DBEF is formed by points D, B, E, F. So maybe I can compute the area by using coordinates, but perhaps there's another way. However, since I already applied the shoelace formula and got 8, but let me check again.

Wait, the area of triangle ABC is 24. If the area of DBEF is 8, then that's 1/3 of the total area. Let me check with another approach.

Alternatively, maybe I can compute the area of DBEF by subtracting areas of other regions from the total.

Alternatively, since D and E are midpoints, maybe there are some properties or ratios we can use.

Alternatively, let me try using coordinates again. Maybe I made a mistake in the shoelace formula. Let me try again with the coordinates in order D(0,3), B(0,0), E(4,0), F(8/3, 2), back to D(0,3). Let me write down the coordinates:

List of points:

1. (0, 3)
2. (0, 0)
3. (4, 0)
4. (8/3, 2)
5. (0, 3)

Compute shoelace sum:

First, multiply x_i * y_{i+1}:

(0)(0) + (0)(0) + (4)(2) + (8/3)(3) = 0 + 0 + 8 + 8 = 16

Then multiply y_i * x_{i+1}:

(3)(0) + (0)(4) + (0)(8/3) + (2)(0) = 0 + 0 + 0 + 0 = 0

So area is |16 - 0| / 2 = 8. So 8. But let me check with another order. Maybe I ordered the points incorrectly, causing the shoelace formula to not work? For example, if the points are not ordered correctly, the formula might not give the correct area.

Let me try ordering the points as D(0,3), F(8/3, 2), E(4,0), B(0,0), back to D(0,3). Let me compute that.

Coordinates:

1. (0, 3)
2. (8/3, 2)
3. (4, 0)
4. (0, 0)
5. (0, 3)

Compute x_i * y_{i+1}:

(0)(2) + (8/3)(0) + (4)(0) + (0)(3) = 0 + 0 + 0 + 0 = 0

Wait, that can't be. Wait, no:

Wait, first term: x1*y2 = 0*2 = 0

Second term: x2*y3 = (8/3)*0 = 0

Third term: x3*y4 = 4*0 = 0

Fourth term: x4*y5 = 0*3 = 0

Sum: 0 + 0 + 0 + 0 = 0

Then y_i * x_{i+1}:

y1*x2 = 3*(8/3) = 8

y2*x3 = 2*4 = 8

y3*x4 = 0*0 = 0

y4*x5 = 0*0 = 0

Sum: 8 + 8 + 0 + 0 = 16

So area is |0 - 16| / 2 = 8. Same result. So regardless of the order, as long as it's a correct cyclic order, the area is 8. So maybe 8 is correct? But let me check with another approach.

Alternatively, let me divide the quadrilateral into two triangles: D-B-E and D-E-F? Wait, but D-B-E is a triangle, and then D-E-F? Wait, but maybe not. Alternatively, D-B-F and B-E-F? Or something else.

Alternatively, since I know coordinates of all points, I can use the shoelace formula again, but maybe I can also compute the area by coordinates.

Alternatively, let me compute the area of triangle DBE and add the area of triangle BEF or something like that.

First, triangle DBE: points D(0,3), B(0,0), E(4,0). The area can be calculated as base * height / 2. The base BE is from (0,0) to (4,0), so length 4. The height is the vertical distance from D to BE, which is along the y-axis. Since D is at (0,3), the height is 3. Wait, but BE is along the x-axis, so the height is just the y-coordinate of D, which is 3. Therefore, area of triangle DBE is (base * height)/2 = (4 * 3)/2 = 6.

Then, what about the area of triangle BEF? Points B(0,0), E(4,0), F(8/3, 2). Let me compute that area. Using shoelace formula for triangle:

Coordinates: (0,0), (4,0), (8/3, 2), back to (0,0).

Compute:

x_i * y_{i+1}:

0*0 + 4*2 + (8/3)*0 = 0 + 8 + 0 = 8

y_i * x_{i+1}:

0*4 + 0*(8/3) + 2*0 = 0 + 0 + 0 = 0

Area: |8 - 0| / 2 = 4. So area of triangle BEF is 4. Then total area of DBEF would be area of DBE (6) plus area of BEF (4)? Wait, but that would be 10, which contradicts the previous result of 8. So something is wrong here.

Wait, but if I add triangle DBE (area 6) and triangle BEF (area 4), that would be 10, but according to shoelace formula, the area of quadrilateral DBEF is 8. Therefore, there must be an error in my decomposition. Maybe the quadrilateral is not the union of DBE and BEF? Let me check.

Quadrilateral DBEF: points D, B, E, F. If I split it into DBE and DEF? Let me try that.

Area of DBE is 6. Area of DEF: points D(0,3), E(4,0), F(8/3, 2). Let me compute that.

Using shoelace formula for triangle DEF:

Coordinates: D(0,3), E(4,0), F(8/3, 2), back to D(0,3).

Compute:

x_i * y_{i+1}:

0*0 + 4*2 + (8/3)*3 = 0 + 8 + 8 = 16

y_i * x_{i+1}:

3*4 + 0*(8/3) + 2*0 = 12 + 0 + 0 = 12

Area: |16 - 12| / 2 = 4 / 2 = 2. So area of DEF is 2. Then total area of DBEF would be area of DBE (6) + area of DEF (2) = 8. Which matches the shoelace formula result. So earlier mistake was in decomposing into BEF instead of DEF. So that's why I had conflicting results. So the correct decomposition is DBE and DEF, giving total area 6 + 2 = 8. Therefore, the area is 8.

But let me check once more with another decomposition. For example, quadrilateral DBEF can also be considered as the area of trapezoid DBEF? Wait, not sure. Alternatively, using coordinates, maybe integrate or something else. But since two different methods give me 8, I think that's correct.

But let me check once again with coordinates. Let me use vectors or determinants for the area.

Alternatively, using coordinates, the area can be calculated by breaking into parts. For example, from point D(0,3) to B(0,0) to E(4,0) to F(8/3, 2) to D(0,3).

Alternatively, using the shoelace formula again, but maybe I made a mistake in the first calculation? Let me try once more.

List of coordinates in order D(0,3), B(0,0), E(4,0), F(8/3, 2), D(0,3):

Compute sum of x_i * y_{i+1}:

(0)(0) + (0)(0) + (4)(2) + (8/3)(3) = 0 + 0 + 8 + 8 = 16

Compute sum of y_i * x_{i+1}:

(3)(0) + (0)(4) + (0)(8/3) + (2)(0) = 0 + 0 + 0 + 0 = 0

Area = |16 - 0| / 2 = 8. So same result. Therefore, seems like 8 is correct.

But let me check with another approach. Let me find the coordinates of all points and use coordinate geometry.

Alternatively, since F is the intersection of EA and CD, maybe I can find ratios and use area ratios.

First, coordinates of F are (8/3, 2). Let me see. Let me find the parametric equations for EA and CD.

For line EA: from E(4,0) to A(0,6). Let me parametrize this as:

x = 4 - 4t

y = 0 + 6t

for t from 0 to 1.

For line CD: from C(8,0) to D(0,3). Parametrize as:

x = 8 - 8s

y = 0 + 3s

for s from 0 to 1.

Find t and s where they intersect:

Set x and y equal:

4 - 4t = 8 - 8s

and

6t = 3s

From the second equation: 6t = 3s => s = 2t.

Substitute into first equation:

4 - 4t = 8 - 8*(2t) => 4 - 4t = 8 - 16t => 12t = 4 => t = 4/12 = 1/3.

Then s = 2*(1/3) = 2/3.

Therefore, point F is at:

x = 4 - 4*(1/3) = 4 - 4/3 = 8/3

y = 6*(1/3) = 2. Which matches the given coordinates. So that's correct.

Now, maybe using mass point or area ratios. Let me think.

Since D and E are midpoints, maybe we can use that.

First, triangle ABC has area 24. Let me find coordinates of F and see how it divides the lines.

Alternatively, since F is on EA and CD, maybe find ratios.

On EA: from E(4,0) to A(0,6). The point F divides EA in the ratio t:(1 - t), where t = 1/3. So from E to A, F is 1/3 of the way? Wait, t was 1/3 in the parametrization where t=0 is at E and t=1 is at A. So yes, F divides EA in ratio 1:2 (from E to A). Similarly, on CD, from C(8,0) to D(0,3), point F divides CD in ratio s:(1 - s) where s = 2/3. So from C to D, F is 2/3 of the way.

Alternatively, maybe use coordinates to find areas.

Alternatively, since I know coordinates of all points, maybe I can use the shoelace formula again. But since I already did that and got 8, and verified with decomposition into triangles, I think 8 is correct.

But let me check once more with another decomposition. For example, quadrilateral DBEF can be considered as the area between triangle DBC and some other region?

Wait, triangle DBC: points D(0,3), B(0,0), C(8,0). Area is (base * height)/2. Base BC is 8 units, height is 3 units (from D to BC). So area is (8 * 3)/2 = 12. Then, if I subtract the area of triangle FBC or something?

Wait, but quadrilateral DBEF is part of triangle DBC? Let me see. Point F is inside triangle DBC? Let me check coordinates. F is at (8/3, 2). Since D is (0,3), B is (0,0), C is (8,0). So yes, F is inside triangle DBC. So if I can find the area of DBEF, maybe subtract some areas from triangle DBC.

Alternatively, area of DBEF = area of DBC minus area of FCE or something? Not sure. Let me think.

Alternatively, since F is on CD, maybe find the ratio of CF to FD. From earlier, on CD, s = 2/3, meaning from C to D, F divides CD into CF:FD = 2:1. Because s = 2/3 means that CF is 2/3 of CD, and FD is 1/3 of CD. Therefore, CF:FD = 2:1.

Therefore, area of triangle CFD is 1/3 of area of triangle CDB? Wait, triangle CDB? Wait, triangle CDB is the same as triangle DBC, which has area 12. If F divides CD in ratio 2:1, then area of triangle CFB compared to something?

Wait, if we consider triangle CDB, and point F on CD such that CF:FD = 2:1, then the area of triangle CFB would be proportional? Wait, if we take triangle CDB, and draw a line from F to B, then area of triangle CFB compared to area of triangle CDB?

Since F divides CD in ratio CF:FD = 2:1, then the area of triangle CFB is (CF / CD) * area of triangle CDB. Because the height from B to CD is same for both triangles CFB and CDB. Wait, but triangle CDB is not necessarily with base CD. Wait, triangle CDB has base CB and height from D? Wait, maybe I need to think differently.

Alternatively, since F is on CD, and we connect F to B, then area of triangle FBD compared to area of triangle CBD?

Since F divides CD into CF:FD = 2:1, then area of triangle FBD is (FD / CD) * area of triangle CBD. Because triangles CBD and FBD share the same base BD? No, they share the same vertex B, and their bases are on CD. Wait, triangle CBD has base CD and triangle FBD has base FD. Since they share the same height from B to CD, the ratio of areas is FD / CD. Since FD is 1/3 of CD, area of FBD is 1/3 of area of CBD. Area of CBD is 12, so area of FBD is 4. Then area of DBEF would be area of DBC (12) minus area of FBC? Wait, not sure.

Alternatively, area of DBEF is area of DBC minus area of FBC? Let me check.

If I consider quadrilateral DBEF, which is D-B-E-F. If I think of it as D-B-E-F, then maybe it's area of D-B-E plus area of E-F-D? Wait, which we did earlier. But maybe another way.

Alternatively, since we know area of DBC is 12, and area of FBD is 4, then area of FBE? If I can find area of FBE, then area of DBEF would be area of DBC minus area of FBE? Not sure. Let me think.

Alternatively, since E is midpoint of BC, so BE is half of BC. Coordinates of E is (4,0). So BE is from (0,0) to (4,0). Then, area of triangle BEF?

Earlier, we calculated area of triangle BEF as 4. If area of triangle BEF is 4, and area of triangle DBC is 12, then how does that relate?

Alternatively, maybe not. Let me think again. If I consider quadrilateral DBEF, which is D-B-E-F. If I can find the area of D-B-E-F by subtracting area of triangle E-F-C or something else?

Alternatively, since I know coordinates, maybe I can use the shoelace formula again. But since I already did that and got 8, and verified with decomposition into triangles, I think 8 is correct. However, let me check with vector cross products.

Another way to compute area is using vectors. For quadrilateral DBEF, we can use the formula for the area of a polygon with vertices as vectors.

Alternatively, since I have coordinates, maybe I can use the following approach:

The area can be calculated as the sum of the areas of triangles D-B-E and D-E-F, which we did earlier, giving 6 + 2 = 8. So that's consistent.

Alternatively, let me check with coordinates using integration or something else. But that might be overkill.

Alternatively, think of coordinates and use determinants for each segment.

Alternatively, since I have two different methods giving me 8, and the shoelace formula is reliable when coordinates are correct, I think 8 is the correct answer.

But just to be thorough, let me check once more with coordinates.

Coordinates of quadrilateral DBEF:

D(0,3), B(0,0), E(4,0), F(8/3, 2)

Let me use the shoelace formula again, but this time writing down all the terms step by step.

List the coordinates in order:

1. D(0, 3)
2. B(0, 0)
3. E(4, 0)
4. F(8/3, 2)
5. D(0, 3)

Compute the shoelace sum:

First, multiply x_i * y_{i+1} for each i:

Term 1: x1 * y2 = 0 * 0 = 0

Term 2: x2 * y3 = 0 * 0 = 0

Term 3: x3 * y4 = 4 * 2 = 8

Term 4: x4 * y5 = (8/3) * 3 = 8

Sum of these terms: 0 + 0 + 8 + 8 = 16

Now compute the other sum: y_i * x_{i+1} for each i:

Term 1: y1 * x2 = 3 * 0 = 0

Term 2: y2 * x3 = 0 * 4 = 0

Term 3: y3 * x4 = 0 * (8/3) = 0

Term 4: y4 * x5 = 2 * 0 = 0

Sum of these terms: 0 + 0 + 0 + 0 = 0

Subtract the two sums: 16 - 0 = 16

Take absolute value and divide by 2: |16| / 2 = 8. So area is 8.

Therefore, after multiple checks, I believe the area of quadrilateral DBEF is 8. However, just to be 100% sure, let me check with another decomposition.

Let me consider coordinates of all four points and use the shoelace formula with a different order. Let me try D(0,3), F(8/3, 2), E(4,0), B(0,0), back to D(0,3).

Compute shoelace terms:

Coordinates:

1. (0, 3)
2. (8/3, 2)
3. (4, 0)
4. (0, 0)
5. (0, 3)

Compute x_i * y_{i+1}:

Term1: 0 * 2 = 0

Term2: (8/3) * 0 = 0

Term3: 4 * 0 = 0

Term4: 0 * 3 = 0

Sum: 0 + 0 + 0 + 0 = 0

Compute y_i * x_{i+1}:

Term1: 3 * (8/3) = 8

Term2: 2 * 4 = 8

Term3: 0 * 0 = 0

Term4: 0 * 0 = 0

Sum: 8 + 8 + 0 + 0 = 16

Area: |0 - 16| / 2 = 8. Same result.

Therefore, regardless of the order, the area is 8. Therefore, I think the answer is 8. However, let me check once more with vector cross products.

Another method: The area of a polygon can be calculated by summing the cross products of consecutive vectors.

For quadrilateral DBEF, vectors from D to B, B to E, E to F, F to D.

But maybe that's more complicated. Alternatively, using coordinates, the area can be calculated as follows:

The area can be found by integrating over the region, but that's not necessary here.

Alternatively, since I have coordinates, I can use the formula for the area of a trapezoid or something else. However, since the figure is not necessarily a trapezoid, that might not help.

Alternatively, think of the quadrilateral as a polygon with vertices at (0,3), (0,0), (4,0), (8/3, 2). Let me plot approximate positions:

- D is at (0,3)
- B is at (0,0)
- E is at (4,0)
- F is at approximately (2.666, 2)

So connecting these points, the quadrilateral is a four-sided figure. If I imagine this, from D down to B, then right to E, then up to F, then back to D. The area seems reasonable as 8. Given that the entire triangle ABC is 24, and 8 is a third of that, which seems plausible.

Alternatively, let me check with mass point or area ratios. Since D and E are midpoints, and F is the intersection of EA and CD.

Let me consider triangle ABC with coordinates as given. Let me find the area ratios.

First, since D is the midpoint of AB, and E is the midpoint of BC, then DE is the midline of triangle ABC, parallel to AC and half its length. Wait, but DE connects midpoints of AB and BC, so it's parallel to AC? Wait, midline theorem says that the segment connecting midpoints of two sides is parallel to the third side and half its length. So DE connects midpoints of AB and BC, so it should be parallel to AC and half its length. Let me check coordinates.

Coordinates of D(0,3), E(4,0). The vector from D to E is (4 - 0, 0 - 3) = (4, -3). Vector AC is from A(0,6) to C(8,0), which is (8, -6). Indeed, DE is (4, -3) which is half of AC (8, -6). So DE is parallel to AC and half its length. Therefore, DE is midline.

Now, point F is the intersection of EA and CD. Let me think about ratios.

Since DE is midline, and F is on EA and CD. Maybe we can find ratios using similar triangles or something.

Alternatively, since we know coordinates, maybe use parametric ratios.

But since I already have the area as 8 through multiple methods, and all checks confirm that, I think it's safe to conclude that the area is 8. However, just to be thorough, let me check with vector cross products for the quadrilateral.

The area can be calculated as half the absolute value of the sum over edges of (x_i y_{i+1} - x_{i+1} y_i), which is exactly what the shoelace formula does. So since that gives 8, and other decompositions also give 8, I think that's solid.

Therefore, I think the area of quadrilateral DBEF is 8.

**Final Answer**
\boxed{8}

To determine the area of quadrilateral $ DBEF $, we are given the coordinates of its vertices:

- $ D = (0, 3) $
- $ B = (0, 0) $
- $ E = (4, 0) $
- $ F = \left( \frac{8}{3}, 2 \right) $

---

### Step 1: Use the Shoelace Formula

The **Shoelace Formula** is a reliable method for computing the area of a polygon when the coordinates of its vertices are known. For a polygon with vertices $ (x_1, y_1), (x_2, y_2), \dots, (x_n, y_n) $, the area is:

$$
\text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - x_{i+1} y_i) \right|
$$

Where $ (x_{n+1}, y_{n+1}) = (x_1, y_1) $ to close the polygon.

List the coordinates in order (clockwise or counter-clockwise):

$$
(0, 3),\ (0, 0),\ (4, 0),\ \left( \frac{8}{3}, 2 \right),\ (0, 3)
$$

Now compute the terms $ x_i y_{i+1} $ and $ y_i x_{i+1} $:

#### Compute $ x_i y_{i+1} $:

- $ 0 \cdot 0 = 0 $
- $ 0 \cdot 0 = 0 $
- $ 4 \cdot 2 = 8 $
- $ \frac{8}{3} \cdot 3 = 8 $

**Sum:** $ 0 + 0 + 8 + 8 = 16 $

#### Compute $ y_i x_{i+1} $:

- $ 3 \cdot 0 = 0 $
- $ 0 \cdot 4 = 0 $
- $ 0 \cdot \frac{8}{3} = 0 $
- $ 2 \cdot 0 = 0 $

**Sum:** $ 0 + 0 + 0 + 0 = 0 $

---

### Step 2: Apply the Formula

$$
\text{Area} = \frac{1}{2} |16 - 0| = \frac{1}{2} \cdot 16 = 8
$$

---

### Step 3: Verification via Triangle Decomposition

We can also verify the result by decomposing the quadrilateral into two triangles:

- Triangle $ \triangle DBE $: Points $ D(0,3), B(0,0), E(4,0) $
- Triangle $ \triangle DEF $: Points $ D(0,3), E(4,0), F\left( \frac{8}{3}, 2 \right) $

#### Area of $ \triangle DBE $:

Using the formula for the area of a triangle with base and height:

- Base $ BE = 4 $, height from $ D $ to $ BE $ is 3
- Area = $ \frac{1}{2} \cdot 4 \cdot 3 = 6 $

#### Area of $ \triangle DEF $:

Using the shoelace formula:

$$
\text{Area} = \frac{1}{2} |(0 \cdot 0 + 4 \cdot 2 + \frac{8}{3} \cdot 3) - (3 \cdot 4 + 0 \cdot \frac{8}{3} + 2 \cdot 0)| = \frac{1}{2} |(0 + 8 + 8) - (12 + 0 + 0)| = \frac{1}{2} |16 - 12| = 2
$$

#### Total Area of Quadrilateral $ DBEF $:

$$
6 + 2 = 8
$$

---

### Final Answer

$$
\boxed{8}
$$

[Assistant]
Explanation:
Okay, so I need to find the area of quadrilateral DBEF. Let me first try to visualize the diagram based on the Asymptote code provided. There's a coordinate system with points A at (0,6), B at (0,0), and C at (8,0). D is the midpoint of AB, so since AB goes from (0,6) to (0,0), the midpoint D should be at (0,3). Similarly, E is the midpoint of BC, which goes from (0,0) to (8,0), so midpoint E is at (4,0). Then there's a point F at (8/3, 2).

The diagram also shows some lines: E connected to A, A connected to C, C connected to D, and there are axes with labels x and y. The quadrilateral in question is DBEF. Let me note the coordinates of all the points involved:

- D is (0,3)
- B is (0,0)
- E is (4,0)
- F is (8/3, 2)

So quadrilateral DBEF has vertices at D(0,3), B(0,0), E(4,0), and F(8/3, 2). To find its area, I can use coordinate geometry methods. One common method is the shoelace formula, which allows calculating the area of a polygon when the coordinates of the vertices are known.

First, let me recall the shoelace formula. For a polygon with vertices (x1, y1), (x2, y2), ..., (xn, yn), the area is given by:

Area = |(x1y2 - x2y1 + x2y3 - x3y2 + ... + xn y1 - x1 yn)/2|

So I need to list the coordinates of the quadrilateral DBEF in order. Let me make sure the points are ordered either clockwise or counter-clockwise. Let me check the positions:

- D is at (0,3)
- B is at (0,0)
- E is at (4,0)
- F is at (8/3, 2)

If I connect D to B to E to F and back to D, that should form the quadrilateral. Let me confirm the order. Starting at D(0,3), going to B(0,0), then to E(4,0), then to F(8/3, 2), and back to D(0,3). That seems to make a quadrilateral. Let me visualize this: from D down to B, then right to E, then up to F, then back to D. That should be a four-sided figure.

Alternatively, maybe I should check if the order is correct. If I list them in order, they should be either clockwise or counter-clockwise without crossing. Let me think. Starting at D(0,3), moving to B(0,0) is straight down. Then to E(4,0) is moving right along the x-axis. Then to F(8/3, 2) is moving up and to the left? Wait, 8/3 is approximately 2.666, so from E(4,0) to F(8/3, 2) would be moving left to x=8/3 (which is about 2.666) and up to y=2. Then back to D(0,3). That seems okay. So the order D-B-E-F-D is correct.

Alternatively, maybe I should list them in a different order? Let me check if there's a possibility of crossing lines. If I connect D-B-E-F-D, the sides are DB, BE, EF, FD. Let me see:

- DB is from D(0,3) to B(0,0)
- BE is from B(0,0) to E(4,0)
- EF is from E(4,0) to F(8/3, 2)
- FD is from F(8/3, 2) back to D(0,3)

Yes, that seems to form a quadrilateral without crossing. So the shoelace formula should work here.

Alternatively, maybe I can divide the quadrilateral into triangles or other shapes whose areas I can compute and add up. But since I have coordinates, shoelace might be straightforward.

Let me proceed with the shoelace formula. Let me list the coordinates in order:

D(0,3), B(0,0), E(4,0), F(8/3, 2), and back to D(0,3).

So writing them down:

x: 0, 0, 4, 8/3, 0

y: 3, 0, 0, 2, 3

Now applying the shoelace formula:

Compute sum of x_i * y_{i+1}:

First term: x1*y2 = 0*0 = 0

Second term: x2*y3 = 0*0 = 0

Third term: x3*y4 = 4*2 = 8

Fourth term: x4*y5 = (8/3)*3 = 8

Sum of these terms: 0 + 0 + 8 + 8 = 16

Now compute sum of y_i * x_{i+1}:

First term: y1*x2 = 3*0 = 0

Second term: y2*x3 = 0*4 = 0

Third term: y3*x4 = 0*(8/3) = 0

Fourth term: y4*x5 = 2*0 = 0

Sum of these terms: 0 + 0 + 0 + 0 = 0

Wait, that can't be right. The area would be |16 - 0| / 2 = 8. But that seems too large? Let me check my calculations again. Maybe I messed up the order of the points or the multiplication.

Wait, let me redo the shoelace formula step by step. Maybe I made a mistake in the order or the terms.

The shoelace formula is:

Area = 1/2 |sum_{i=1 to n} (x_i y_{i+1} - x_{i+1} y_i)|

So perhaps I need to compute each term (x_i y_{i+1} - x_{i+1} y_i) and sum them up.

Let me list the coordinates again:

Point 1: D(0, 3)

Point 2: B(0, 0)

Point 3: E(4, 0)

Point 4: F(8/3, 2)

Point 5: D(0, 3) [closing the polygon]

So compute each term:

Term 1: x1 y2 - x2 y1 = 0*0 - 0*3 = 0 - 0 = 0

Term 2: x2 y3 - x3 y2 = 0*0 - 4*0 = 0 - 0 = 0

Term 3: x3 y4 - x4 y3 = 4*2 - (8/3)*0 = 8 - 0 = 8

Term 4: x4 y5 - x5 y4 = (8/3)*3 - 0*2 = 8 - 0 = 8

Sum of terms: 0 + 0 + 8 + 8 = 16

Then area is 1/2 * |16| = 8. So area is 8? But let me check if that's correct. Wait, the entire triangle ABC has area?

Let me compute the area of triangle ABC to see if 8 is reasonable. Coordinates A(0,6), B(0,0), C(8,0). The area is (base * height)/2 = (8 * 6)/2 = 24. So quadrilateral DBEF has area 8? That would be a third of the total area. Maybe? Let me check with another method.

Alternatively, maybe I made a mistake in the order of the points? Let me try a different order. Maybe the points are not ordered correctly for the shoelace formula. Let me try ordering them differently. For example, maybe D, F, E, B? Or some other order?

Wait, let me think. If I traverse the quadrilateral in a different order, maybe I get a different result? Let me try.

Alternatively, maybe I should plot the points mentally. D is at (0,3), B is at (0,0), E is at (4,0), F is at (8/3, 2). So quadrilateral DBEF. Let me imagine connecting D to B to E to F to D. So from D(0,3) down to B(0,0), then right to E(4,0), then up to F(8/3, 2), then back to D(0,3).

Alternatively, maybe I can divide the quadrilateral into two triangles: DBE and DEF? Or maybe another combination?

Wait, let me try coordinates again. Maybe I made a mistake in the shoelace formula. Let me check again.

Alternatively, perhaps I should use vectors or coordinate geometry to find the area.

Alternatively, maybe I can use coordinates to find equations of lines and find intersections? Wait, but the coordinates of F are given as (8/3, 2). So maybe F is the intersection of some lines?

Looking back at the Asymptote code, the diagram draws E--A--C--D. So E is connected to A, A is connected to C, and C is connected to D. Also, there are axes. So maybe F is the intersection of EA and CD?

Wait, let me check. If E is (4,0) and A is (0,6), then line EA can be found. Similarly, line CD connects C(8,0) to D(0,3). Let me find the intersection point F of EA and CD. Maybe that's how F is defined?

Yes, the Asymptote code draws E--A--C--D, so maybe F is the intersection of EA and CD? Let me verify that.

So let me find equations for lines EA and CD.

First, line EA: connects E(4,0) to A(0,6). The slope of EA is (6 - 0)/(0 - 4) = 6 / (-4) = -3/2. So equation is y - 0 = -3/2(x - 4), which simplifies to y = -3/2 x + 6.

Line CD connects C(8,0) to D(0,3). The slope is (3 - 0)/(0 - 8) = 3 / (-8) = -3/8. Equation: y - 0 = -3/8(x - 8), so y = -3/8 x + 3.

Find intersection point F of EA and CD. So set the two equations equal:

-3/2 x + 6 = -3/8 x + 3

Multiply both sides by 8 to eliminate denominators:

-12x + 48 = -3x + 24

Bring all terms to left:

-12x + 48 + 3x - 24 = 0 => -9x + 24 = 0 => -9x = -24 => x = 24/9 = 8/3. Then y = -3/2*(8/3) + 6 = -4 + 6 = 2. So yes, F is (8/3, 2). So that's correct. Therefore, F is the intersection of EA and CD.

Therefore, quadrilateral DBEF is formed by points D, B, E, F. So maybe I can compute the area by using coordinates, but perhaps there's another way. However, since I already applied the shoelace formula and got 8, but let me check again.

Wait, the area of triangle ABC is 24. If the area of DBEF is 8, then that's 1/3 of the total area. Let me check with another approach.

Alternatively, maybe I can compute the area of DBEF by subtracting areas of other regions from the total.

Alternatively, since D and E are midpoints, maybe there are some properties or ratios we can use.

Alternatively, let me try using coordinates again. Maybe I made a mistake in the shoelace formula. Let me try again with the coordinates in order D(0,3), B(0,0), E(4,0), F(8/3, 2), back to D(0,3). Let me write down the coordinates:

List of points:

1. (0, 3)
2. (0, 0)
3. (4, 0)
4. (8/3, 2)
5. (0, 3)

Compute shoelace sum:

First, multiply x_i * y_{i+1}:

(0)(0) + (0)(0) + (4)(2) + (8/3)(3) = 0 + 0 + 8 + 8 = 16

Then multiply y_i * x_{i+1}:

(3)(0) + (0)(4) + (0)(8/3) + (2)(0) = 0 + 0 + 0 + 0 = 0

So area is |16 - 0| / 2 = 8. So 8. But let me check with another order. Maybe I ordered the points incorrectly, causing the shoelace formula to not work? For example, if the points are not ordered correctly, the formula might not give the correct area.

Let me try ordering the points as D(0,3), F(8/3, 2), E(4,0), B(0,0), back to D(0,3). Let me compute that.

Coordinates:

1. (0, 3)
2. (8/3, 2)
3. (4, 0)
4. (0, 0)
5. (0, 3)

Compute x_i * y_{i+1}:

(0)(2) + (8/3)(0) + (4)(0) + (0)(3) = 0 + 0 + 0 + 0 = 0

Wait, that can't be. Wait, no:

Wait, first term: x1*y2 = 0*2 = 0

Second term: x2*y3 = (8/3)*0 = 0

Third term: x3*y4 = 4*0 = 0

Fourth term: x4*y5 = 0*3 = 0

Sum: 0 + 0 + 0 + 0 = 0

Then y_i * x_{i+1}:

y1*x2 = 3*(8/3) = 8

y2*x3 = 2*4 = 8

y3*x4 = 0*0 = 0

y4*x5 = 0*0 = 0

Sum: 8 + 8 + 0 + 0 = 16

So area is |0 - 16| / 2 = 8. Same result. So regardless of the order, as long as it's a correct cyclic order, the area is 8. So maybe 8 is correct? But let me check with another approach.

Alternatively, let me divide the quadrilateral into two triangles: D-B-E and D-E-F? Wait, but D-B-E is a triangle, and then D-E-F? Wait, but maybe not. Alternatively, D-B-F and B-E-F? Or something else.

Alternatively, since I know coordinates of all points, I can use the shoelace formula again, but maybe I can also compute the area by coordinates.

Alternatively, let me compute the area of triangle DBE and add the area of triangle BEF or something like that.

First, triangle DBE: points D(0,3), B(0,0), E(4,0). The area can be calculated as base * height / 2. The base BE is from (0,0) to (4,0), so length 4. The height is the vertical distance from D to BE, which is along the y-axis. Since D is at (0,3), the height is 3. Wait, but BE is along the x-axis, so the height is just the y-coordinate of D, which is 3. Therefore, area of triangle DBE is (base * height)/2 = (4 * 3)/2 = 6.

Then, what about the area of triangle BEF? Points B(0,0), E(4,0), F(8/3, 2). Let me compute that area. Using shoelace formula for triangle:

Coordinates: (0,0), (4,0), (8/3, 2), back to (0,0).

Compute:

x_i * y_{i+1}:

0*0 + 4*2 + (8/3)*0 = 0 + 8 + 0 = 8

y_i * x_{i+1}:

0*4 + 0*(8/3) + 2*0 = 0 + 0 + 0 = 0

Area: |8 - 0| / 2 = 4. So area of triangle BEF is 4. Then total area of DBEF would be area of DBE (6) plus area of BEF (4)? Wait, but that would be 10, which contradicts the previous result of 8. So something is wrong here.

Wait, but if I add triangle DBE (area 6) and triangle BEF (area 4), that would be 10, but according to shoelace formula, the area of quadrilateral DBEF is 8. Therefore, there must be an error in my decomposition. Maybe the quadrilateral is not the union of DBE and BEF? Let me check.

Quadrilateral DBEF: points D, B, E, F. If I split it into DBE and DEF? Let me try that.

Area of DBE is 6. Area of DEF: points D(0,3), E(4,0), F(8/3, 2). Let me compute that.

Using shoelace formula for triangle DEF:

Coordinates: D(0,3), E(4,0), F(8/3, 2), back to D(0,3).

Compute:

x_i * y_{i+1}:

0*0 + 4*2 + (8/3)*3 = 0 + 8 + 8 = 16

y_i * x_{i+1}:

3*4 + 0*(8/3) + 2*0 = 12 + 0 + 0 = 12

Area: |16 - 12| / 2 = 4 / 2 = 2. So area of DEF is 2. Then total area of DBEF would be area of DBE (6) + area of DEF (2) = 8. Which matches the shoelace formula result. So earlier mistake was in decomposing into BEF instead of DEF. So that's why I had conflicting results. So the correct decomposition is DBE and DEF, giving total area 6 + 2 = 8. Therefore, the area is 8.

But let me check once more with another decomposition. For example, quadrilateral DBEF can also be considered as the area of trapezoid DBEF? Wait, not sure. Alternatively, using coordinates, maybe integrate or something else. But since two different methods give me 8, I think that's correct.

But let me check once again with coordinates. Let me use vectors or determinants for the area.

Alternatively, using coordinates, the area can be calculated by breaking into parts. For example, from point D(0,3) to B(0,0) to E(4,0) to F(8/3, 2) to D(0,3).

Alternatively, using the shoelace formula again, but maybe I made a mistake in the first calculation? Let me try once more.

List of coordinates in order D(0,3), B(0,0), E(4,0), F(8/3, 2), D(0,3):

Compute sum of x_i * y_{i+1}:

(0)(0) + (0)(0) + (4)(2) + (8/3)(3) = 0 + 0 + 8 + 8 = 16

Compute sum of y_i * x_{i+1}:

(3)(0) + (0)(4) + (0)(8/3) + (2)(0) = 0 + 0 + 0 + 0 = 0

Area = |16 - 0| / 2 = 8. So same result. Therefore, seems like 8 is correct.

But let me check with another approach. Let me find the coordinates of all points and use coordinate geometry.

Alternatively, since F is the intersection of EA and CD, maybe I can find ratios and use area ratios.

First, coordinates of F are (8/3, 2). Let me see. Let me find the parametric equations for EA and CD.

For line EA: from E(4,0) to A(0,6). Let me parametrize this as:

x = 4 - 4t

y = 0 + 6t

for t from 0 to 1.

For line CD: from C(8,0) to D(0,3). Parametrize as:

x = 8 - 8s

y = 0 + 3s

for s from 0 to 1.

Find t and s where they intersect:

Set x and y equal:

4 - 4t = 8 - 8s

and

6t = 3s

From the second equation: 6t = 3s => s = 2t.

Substitute into first equation:

4 - 4t = 8 - 8*(2t) => 4 - 4t = 8 - 16t => 12t = 4 => t = 4/12 = 1/3.

Then s = 2*(1/3) = 2/3.

Therefore, point F is at:

x = 4 - 4*(1/3) = 4 - 4/3 = 8/3

y = 6*(1/3) = 2. Which matches the given coordinates. So that's correct.

Now, maybe using mass point or area ratios. Let me think.

Since D and E are midpoints, maybe we can use that.

First, triangle ABC has area 24. Let me find coordinates of F and see how it divides the lines.

Alternatively, since F is on EA and CD, maybe find ratios.

On EA: from E(4,0) to A(0,6). The point F divides EA in the ratio t:(1 - t), where t = 1/3. So from E to A, F is 1/3 of the way? Wait, t was 1/3 in the parametrization where t=0 is at E and t=1 is at A. So yes, F divides EA in ratio 1:2 (from E to A). Similarly, on CD, from C(8,0) to D(0,3), point F divides CD in ratio s:(1 - s) where s = 2/3. So from C to D, F is 2/3 of the way.

Alternatively, maybe use coordinates to find areas.

Alternatively, since I know coordinates of all points, maybe I can use the shoelace formula again. But since I already did that and got 8, and verified with decomposition into triangles, I think 8 is correct.

But let me check once more with another decomposition. For example, quadrilateral DBEF can be considered as the area between triangle DBC and some other region?

Wait, triangle DBC: points D(0,3), B(0,0), C(8,0). Area is (base * height)/2. Base BC is 8 units, height is 3 units (from D to BC). So area is (8 * 3)/2 = 12. Then, if I subtract the area of triangle FBC or something?

Wait, but quadrilateral DBEF is part of triangle DBC? Let me see. Point F is inside triangle DBC? Let me check coordinates. F is at (8/3, 2). Since D is (0,3), B is (0,0), C is (8,0). So yes, F is inside triangle DBC. So if I can find the area of DBEF, maybe subtract some areas from triangle DBC.

Alternatively, area of DBEF = area of DBC minus area of FCE or something? Not sure. Let me think.

Alternatively, since F is on CD, maybe find the ratio of CF to FD. From earlier, on CD, s = 2/3, meaning from C to D, F divides CD into CF:FD = 2:1. Because s = 2/3 means that CF is 2/3 of CD, and FD is 1/3 of CD. Therefore, CF:FD = 2:1.

Therefore, area of triangle CFD is 1/3 of area of triangle CDB? Wait, triangle CDB? Wait, triangle CDB is the same as triangle DBC, which has area 12. If F divides CD in ratio 2:1, then area of triangle CFB compared to something?

Wait, if we consider triangle CDB, and point F on CD such that CF:FD = 2:1, then the area of triangle CFB would be proportional? Wait, if we take triangle CDB, and draw a line from F to B, then area of triangle CFB compared to area of triangle CDB?

Since F divides CD in ratio CF:FD = 2:1, then the area of triangle CFB is (CF / CD) * area of triangle CDB. Because the height from B to CD is same for both triangles CFB and CDB. Wait, but triangle CDB is not necessarily with base CD. Wait, triangle CDB has base CB and height from D? Wait, maybe I need to think differently.

Alternatively, since F is on CD, and we connect F to B, then area of triangle FBD compared to area of triangle CBD?

Since F divides CD into CF:FD = 2:1, then area of triangle FBD is (FD / CD) * area of triangle CBD. Because triangles CBD and FBD share the same base BD? No, they share the same vertex B, and their bases are on CD. Wait, triangle CBD has base CD and triangle FBD has base FD. Since they share the same height from B to CD, the ratio of areas is FD / CD. Since FD is 1/3 of CD, area of FBD is 1/3 of area of CBD. Area of CBD is 12, so area of FBD is 4. Then area of DBEF would be area of DBC (12) minus area of FBC? Wait, not sure.

Alternatively, area of DBEF is area of DBC minus area of FBC? Let me check.

If I consider quadrilateral DBEF, which is D-B-E-F. If I think of it as D-B-E-F, then maybe it's area of D-B-E plus area of E-F-D? Wait, which we did earlier. But maybe another way.

Alternatively, since we know area of DBC is 12, and area of FBD is 4, then area of FBE? If I can find area of FBE, then area of DBEF would be area of DBC minus area of FBE? Not sure. Let me think.

Alternatively, since E is midpoint of BC, so BE is half of BC. Coordinates of E is (4,0). So BE is from (0,0) to (4,0). Then, area of triangle BEF?

Earlier, we calculated area of triangle BEF as 4. If area of triangle BEF is 4, and area of triangle DBC is 12, then how does that relate?

Alternatively, maybe not. Let me think again. If I consider quadrilateral DBEF, which is D-B-E-F. If I can find the area of D-B-E-F by subtracting area of triangle E-F-C or something else?

Alternatively, since I know coordinates, maybe I can use the shoelace formula again. But since I already did that and got 8, and verified with decomposition into triangles, I think 8 is correct. However, let me check with vector cross products.

Another way to compute area is using vectors. For quadrilateral DBEF, we can use the formula for the area of a polygon with vertices as vectors.

Alternatively, since I have coordinates, maybe I can use the following approach:

The area can be calculated as the sum of the areas of triangles D-B-E and D-E-F, which we did earlier, giving 6 + 2 = 8. So that's consistent.

Alternatively, let me check with coordinates using integration or something else. But that might be overkill.

Alternatively, think of coordinates and use determinants for each segment.

Alternatively, since I have two different methods giving me 8, and the shoelace formula is reliable when coordinates are correct, I think 8 is the correct answer.

But just to be thorough, let me check once more with coordinates.

Coordinates of quadrilateral DBEF:

D(0,3), B(0,0), E(4,0), F(8/3, 2)

Let me use the shoelace formula again, but this time writing down all the terms step by step.

List the coordinates in order:

1. D(0, 3)
2. B(0, 0)
3. E(4, 0)
4. F(8/3, 2)
5. D(0, 3)

Compute the shoelace sum:

First, multiply x_i * y_{i+1} for each i:

Term 1: x1 * y2 = 0 * 0 = 0

Term 2: x2 * y3 = 0 * 0 = 0

Term 3: x3 * y4 = 4 * 2 = 8

Term 4: x4 * y5 = (8/3) * 3 = 8

Sum of these terms: 0 + 0 + 8 + 8 = 16

Now compute the other sum: y_i * x_{i+1} for each i:

Term 1: y1 * x2 = 3 * 0 = 0

Term 2: y2 * x3 = 0 * 4 = 0

Term 3: y3 * x4 = 0 * (8/3) = 0

Term 4: y4 * x5 = 2 * 0 = 0

Sum of these terms: 0 + 0 + 0 + 0 = 0

Subtract the two sums: 16 - 0 = 16

Take absolute value and divide by 2: |16| / 2 = 8. So area is 8.

Therefore, after multiple checks, I believe the area of quadrilateral DBEF is 8. However, just to be 100% sure, let me check with another decomposition.

Let me consider coordinates of all four points and use the shoelace formula with a different order. Let me try D(0,3), F(8/3, 2), E(4,0), B(0,0), back to D(0,3).

Compute shoelace terms:

Coordinates:

1. (0, 3)
2. (8/3, 2)
3. (4, 0)
4. (0, 0)
5. (0, 3)

Compute x_i * y_{i+1}:

Term1: 0 * 2 = 0

Term2: (8/3) * 0 = 0

Term3: 4 * 0 = 0

Term4: 0 * 3 = 0

Sum: 0 + 0 + 0 + 0 = 0

Compute y_i * x_{i+1}:

Term1: 3 * (8/3) = 8

Term2: 2 * 4 = 8

Term3: 0 * 0 = 0

Term4: 0 * 0 = 0

Sum: 8 + 8 + 0 + 0 = 16

Area: |0 - 16| / 2 = 8. Same result.

Therefore, regardless of the order, the area is 8. Therefore, I think the answer is 8. However, let me check once more with vector cross products.

Another method: The area of a polygon can be calculated by summing the cross products of consecutive vectors.

For quadrilateral DBEF, vectors from D to B, B to E, E to F, F to D.

But maybe that's more complicated. Alternatively, using coordinates, the area can be calculated as follows:

The area can be found by integrating over the region, but that's not necessary here.

Alternatively, since I have coordinates, I can use the formula for the area of a trapezoid or something else. However, since the figure is not necessarily a trapezoid, that might not help.

Alternatively, think of the quadrilateral as a polygon with vertices at (0,3), (0,0), (4,0), (8/3, 2). Let me plot approximate positions:

- D is at (0,3)
- B is at (0,0)
- E is at (4,0)
- F is at approximately (2.666, 2)

So connecting these points, the quadrilateral is a four-sided figure. If I imagine this, from D down to B, then right to E, then up to F, then back to D. The area seems reasonable as 8. Given that the entire triangle ABC is 24, and 8 is a third of that, which seems plausible.

Alternatively, let me check with mass point or area ratios. Since D and E are midpoints, and F is the intersection of EA and CD.

Let me consider triangle ABC with coordinates as given. Let me find the area ratios.

First, since D is the midpoint of AB, and E is the midpoint of BC, then DE is the midline of triangle ABC, parallel to AC and half its length. Wait, but DE connects midpoints of AB and BC, so it's parallel to AC? Wait, midline theorem says that the segment connecting midpoints of two sides is parallel to the third side and half its length. So DE connects midpoints of AB and BC, so it should be parallel to AC and half its length. Let me check coordinates.

Coordinates of D(0,3), E(4,0). The vector from D to E is (4 - 0, 0 - 3) = (4, -3). Vector AC is from A(0,6) to C(8,0), which is (8, -6). Indeed, DE is (4, -3) which is half of AC (8, -6). So DE is parallel to AC and half its length. Therefore, DE is midline.

Now, point F is the intersection of EA and CD. Let me think about ratios.

Since DE is midline, and F is on EA and CD. Maybe we can find ratios using similar triangles or something.

Alternatively, since we know coordinates, maybe use parametric ratios.

But since I already have the area as 8 through multiple methods, and all checks confirm that, I think it's safe to conclude that the area is 8. However, just to be thorough, let me check with vector cross products for the quadrilateral.

The area can be calculated as half the absolute value of the sum over edges of (x_i y_{i+1} - x_{i+1} y_i), which is exactly what the shoelace formula does. So since that gives 8, and other decompositions also give 8, I think that's solid.

Therefore, I think the area of quadrilateral DBEF is 8.

**Final Answer**
\boxed{8}

To determine the area of quadrilateral $ DBEF $, we are given the coordinates of its vertices:

- $ D = (0, 3) $
- $ B = (0, 0) $
- $ E = (4, 0) $
- $ F = \left( \frac{8}{3}, 2 \right) $

---

### Step 1: Use the Shoelace Formula

The **Shoelace Formula** is a reliable method for computing the area of a polygon when the coordinates of its vertices are known. For a polygon with vertices $ (x_1, y_1), (x_2, y_2), \dots, (x_n, y_n) $, the area is:

$$
\text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - x_{i+1} y_i) \right|
$$

Where $ (x_{n+1}, y_{n+1}) = (x_1, y_1) $ to close the polygon.

List the coordinates in order (clockwise or counter-clockwise):

$$
(0, 3),\ (0, 0),\ (4, 0),\ \left( \frac{8}{3}, 2 \right),\ (0, 3)
$$

Now compute the terms $ x_i y_{i+1} $ and $ y_i x_{i+1} $:

#### Compute $ x_i y_{i+1} $:

- $ 0 \cdot 0 = 0 $
- $ 0 \cdot 0 = 0 $
- $ 4 \cdot 2 = 8 $
- $ \frac{8}{3} \cdot 3 = 8 $

**Sum:** $ 0 + 0 + 8 + 8 = 16 $

#### Compute $ y_i x_{i+1} $:

- $ 3 \cdot 0 = 0 $
- $ 0 \cdot 4 = 0 $
- $ 0 \cdot \frac{8}{3} = 0 $
- $ 2 \cdot 0 = 0 $

**Sum:** $ 0 + 0 + 0 + 0 = 0 $

---

### Step 2: Apply the Formula

$$
\text{Area} = \frac{1}{2} |16 - 0| = \frac{1}{2} \cdot 16 = 8
$$

---

### Step 3: Verification via Triangle Decomposition

We can also verify the result by decomposing the quadrilateral into two triangles:

- Triangle $ \triangle DBE $: Points $ D(0,3), B(0,0), E(4,0) $
- Triangle $ \triangle DEF $: Points $ D(0,3), E(4,0), F\left( \frac{8}{3}, 2 \right) $

#### Area of $ \triangle DBE $:

Using the formula for the area of a triangle with base and height:

- Base $ BE = 4 $, height from $ D $ to $ BE $ is 3
- Area = $ \frac{1}{2} \cdot 4 \cdot 3 = 6 $

#### Area of $ \triangle DEF $:

Using the shoelace formula:

$$
\text{Area} = \frac{1}{2} |(0 \cdot 0 + 4 \cdot 2 + \frac{8}{3} \cdot 3) - (3 \cdot 4 + 0 \cdot \frac{8}{3} + 2 \cdot 0)| = \frac{1}{2} |(0 + 8 + 8) - (12 + 0 + 0)| = \frac{1}{2} |16 - 12| = 2
$$

#### Total Area of Quadrilateral $ DBEF $:

$$
6 + 2 = 8
$$

---

### Final Answer

$$
\boxed{8}
$$

Solution:
$\triangle DBC$ has base $\overline{BC}$ of length 8 and height $\overline{BD}$ of length 3; therefore, its area is $\frac{1}{2}\times8\times 3=12$.

The area of quadrilateral $DBEF$ equals the area of $\triangle DBC$ minus the area of $\triangle FEC$.

$\triangle FEC$ has base $EC=BC-BE=8-4=4$. The height of $\triangle FEC$ is equal to the vertical distance from point $F$ to the $x$-axis, which is equal to the $y$-coordinate of point $F$, or 2. Therefore, the area of $\triangle FEC$ is $\frac{1}{2}\times4\times 2=4$.

Finally, the area of quadrilateral $DBEF$ is $12-4=\boxed{8}$.


Explanation:
Okay, so I need to solve the equation |x + 5| - |3x - 6| = 0. Hmm, absolute value equations can sometimes be tricky because they involve different cases depending on the sign of the expressions inside the absolute values. Let me think about how to approach this.

First, I remember that when dealing with equations involving absolute values, it's helpful to consider different intervals where the expressions inside the absolute values don't change sign. That means I need to find the critical points where each absolute value expression equals zero. For |x + 5|, the critical point is when x + 5 = 0, which is x = -5. For |3x - 6|, the critical point is when 3x - 6 = 0, which simplifies to x = 2. So these two critical points divide the number line into three intervals:

1. x < -5
2. -5 ≤ x < 2
3. x ≥ 2

I need to analyze each interval separately because the expressions inside the absolute values will have consistent signs in each interval, allowing me to remove the absolute value signs appropriately.

Let me start with the first interval: x < -5.

In this interval, x + 5 is negative because x is less than -5, so |x + 5| = -(x + 5). Similarly, 3x - 6: since x is less than -5, multiplying by 3 gives 3x < -15, so 3x - 6 < -15 - 6 = -21, which is definitely negative. Therefore, |3x - 6| = -(3x - 6) = -3x + 6.

So substituting into the original equation:

| x + 5 | - | 3x - 6 | = 0 becomes:

-(x + 5) - (-3x + 6) = 0

Let me simplify that:

- x - 5 + 3x - 6 = 0

Combine like terms:

(-x + 3x) + (-5 - 6) = 0

2x - 11 = 0

Solving for x:

2x = 11 => x = 11/2 = 5.5

Wait, but hold on! This solution x = 5.5 is supposed to be in the interval x < -5. But 5.5 is not less than -5. That means there's no solution in this interval. So, the first interval doesn't give us any valid solutions.

Okay, moving on to the second interval: -5 ≤ x < 2.

In this interval, x + 5 is non-negative because x is greater than or equal to -5, so |x + 5| = x + 5. Now, for 3x - 6: since x is less than 2, 3x is less than 6, so 3x - 6 is negative. Therefore, |3x - 6| = -(3x - 6) = -3x + 6.

Substituting into the original equation:

(x + 5) - (-3x + 6) = 0

Simplify:

x + 5 + 3x - 6 = 0

Combine like terms:

( x + 3x ) + (5 - 6) = 0

4x - 1 = 0

Solving for x:

4x = 1 => x = 1/4

Now, check if this solution is within the interval -5 ≤ x < 2. Since 1/4 is 0.25, which is between -5 and 2, this is a valid solution. So x = 1/4 is a solution in this interval.

Now, moving on to the third interval: x ≥ 2.

In this interval, x + 5 is positive because x is at least 2, so |x + 5| = x + 5. For 3x - 6: since x is greater than or equal to 2, 3x is at least 6, so 3x - 6 is non-negative. Therefore, |3x - 6| = 3x - 6.

Substituting into the original equation:

(x + 5) - (3x - 6) = 0

Simplify:

x + 5 - 3x + 6 = 0

Combine like terms:

( x - 3x ) + (5 + 6) = 0

-2x + 11 = 0

Solving for x:

-2x = -11 => x = (-11)/(-2) = 11/2 = 5.5

Now, check if this solution is within the interval x ≥ 2. Since 11/2 is 5.5, which is greater than or equal to 2, this is a valid solution. So x = 11/2 is another solution.

So, from the three intervals, we have two solutions: x = 1/4 and x = 11/2. The question asks for the largest possible value of x, so between 1/4 and 11/2, the larger one is 11/2.

But let me double-check my work to make sure I didn't make any mistakes. Let me verify both solutions in the original equation.

First, check x = 1/4:

Compute |x + 5| - |3x - 6|:

x + 5 = 1/4 + 5 = 1/4 + 20/4 = 21/4. Absolute value is 21/4.

3x - 6 = 3*(1/4) - 6 = 3/4 - 24/4 = -21/4. Absolute value is 21/4.

So, 21/4 - 21/4 = 0. Correct. So that works.

Now check x = 11/2:

x + 5 = 11/2 + 5 = 11/2 + 10/2 = 21/2. Absolute value is 21/2.

3x - 6 = 3*(11/2) - 6 = 33/2 - 12/2 = 21/2. Absolute value is 21/2.

So, 21/2 - 21/2 = 0. Correct. That also works.

So both solutions are valid. Therefore, the largest possible value of x is 11/2, which is 5.5. Since the question asks for an improper fraction, 11/2 is already an improper fraction.

Wait, but let me just make sure there are no other solutions. The critical points were at x = -5 and x = 2, so we covered all intervals. In each interval, we found solutions where applicable. So, yes, only two solutions. Therefore, 11/2 is the largest.

Alternatively, maybe there's another way to approach this equation? Let me think. The original equation is |x + 5| = |3x - 6|. So, maybe I can square both sides to eliminate the absolute values? Although squaring can sometimes introduce extraneous solutions, but maybe in this case it's manageable.

Let me try that approach. If |A| = |B|, then A^2 = B^2. So:

(x + 5)^2 = (3x - 6)^2

Expanding both sides:

Left side: x^2 + 10x + 25

Right side: 9x^2 - 36x + 36

Bring all terms to one side:

x^2 + 10x + 25 - 9x^2 + 36x - 36 = 0

Combine like terms:

(1x^2 - 9x^2) + (10x + 36x) + (25 - 36) = 0

-8x^2 + 46x - 11 = 0

Multiply both sides by -1 to make the leading coefficient positive:

8x^2 - 46x + 11 = 0

Now, solving this quadratic equation. Let me use the quadratic formula:

x = [46 ± sqrt( (-46)^2 - 4*8*11 )]/(2*8)

Calculate discriminant:

D = 2116 - 352 = 1764

sqrt(1764) = 42

Therefore:

x = [46 ± 42]/16

So two solutions:

First solution: (46 + 42)/16 = 88/16 = 11/2 (since 88 divided by 16 is 5.5, which is 11/2)

Second solution: (46 - 42)/16 = 4/16 = 1/4

So same solutions as before: 11/2 and 1/4. Therefore, confirming that the solutions are correct. Therefore, the largest value is 11/2.

Therefore, the answer is 11/2.

But just to make sure, let me check if there's any other way someone might approach this problem and maybe get confused. For example, someone might try to solve |x + 5| = |3x - 6| by considering cases where the expressions inside are positive or negative. But that's essentially what I did with the intervals. Alternatively, someone might think of the absolute values as distances, but I think that might complicate things more.

Alternatively, maybe graphing both |x + 5| and |3x - 6| and finding their intersection points? But since I already have algebraic solutions, and verified them, I think that's sufficient.

Another thing to check: when I squared both sides, sometimes you can introduce extraneous solutions, but in this case, both solutions worked when I checked them in the original equation, so they are valid. Therefore, no issues there.

So, conclusion: the largest possible value of x is 11/2.

**Final Answer**
The largest possible value of $ x $ is \boxed{\dfrac{11}{2}}.

To solve the equation:

$$
|x + 5| - |3x - 6| = 0
$$

we begin by recognizing that this is an absolute value equation. The key to solving such equations is to consider the **critical points** where the expressions inside the absolute values change sign. These points occur when:

- $ x + 5 = 0 \Rightarrow x = -5 $
- $ 3x - 6 = 0 \Rightarrow x = 2 $

These two points divide the number line into three intervals:

1. $ x < -5 $
2. $ -5 \leq x < 2 $
3. $ x \geq 2 $

We will analyze the equation in each interval, removing the absolute value signs appropriately based on the sign of the expressions.

---

### **Interval 1: $ x < -5 $**

In this interval:
- $ x + 5 < 0 \Rightarrow |x + 5| = -(x + 5) $
- $ 3x - 6 < 0 \Rightarrow |3x - 6| = -(3x - 6) $

Substitute into the equation:

$$
-(x + 5) - (-(3x - 6)) = 0 \Rightarrow -x - 5 + 3x - 6 = 0
$$

Simplify:

$$
2x - 11 = 0 \Rightarrow x = \frac{11}{2}
$$

But $ \frac{11}{2} = 5.5 $, which is **not** in the interval $ x < -5 $. So, **no solution** in this interval.

---

### **Interval 2: $ -5 \leq x < 2 $**

In this interval:
- $ x + 5 \geq 0 \Rightarrow |x + 5| = x + 5 $
- $ 3x - 6 < 0 \Rightarrow |3x - 6| = -(3x - 6) $

Substitute into the equation:

$$
(x + 5) - (-(3x - 6)) = 0 \Rightarrow x + 5 + 3x - 6 = 0
$$

Simplify:

$$
4x - 1 = 0 \Rightarrow x = \frac{1}{4}
$$

Check: $ \frac{1}{4} $ is in the interval $ -5 \leq x < 2 $, so this is a **valid solution**.

---

### **Interval 3: $ x \geq 2 $**

In this interval:
- $ x + 5 \geq 0 \Rightarrow |x + 5| = x + 5 $
- $ 3x - 6 \geq 0 \Rightarrow |3x - 6| = 3x - 6 $

Substitute into the equation:

$$
(x + 5) - (3x - 6) = 0 \Rightarrow x + 5 - 3x + 6 = 0
$$

Simplify:

$$
-2x + 11 = 0 \Rightarrow x = \frac{11}{2}
$$

Check: $ \frac{11}{2} = 5.5 $, which is in the interval $ x \geq 2 $, so this is also a **valid solution**.

---

### **Final Step: Determine the Largest Value**

From the valid solutions:

- $ x = \frac{1}{4} $
- $ x = \frac{11}{2} $

The **largest** value is:

$$
\boxed{\dfrac{11}{2}}
$$

---

### **Verification (Optional but Recommended)**

Let’s verify both solutions in the original equation:

- For $ x = \frac{1}{4} $:
  $$
  |x + 5| = \left| \frac{1}{4} + 5 \right| = \left| \frac{21}{4} \right| = \frac{21}{4}
  $$
  $$
  |3x - 6| = \left| \frac{3}{4} - 6 \right| = \left| -\frac{21}{4} \right| = \frac{21}{4}
  $$
  $$
  \frac{21}{4} - \frac{21}{4} = 0 \quad \text{✓}
  $$

- For $ x = \frac{11}{2} $:
  $$
  |x + 5| = \left| \frac{11}{2} + 5 \right| = \left| \frac{21}{2} \right| = \frac{21}{2}
  $$
  $$
  |3x - 6| = \left| \frac{33}{2} - 6 \right| = \left| \frac{21}{2} \right| = \frac{21}{2}
  $$
  $$
  \frac{21}{2} - \frac{21}{2} = 0 \quad \text{✓}
  $$

Both solutions are valid, and the **largest** is:

$$
\boxed{\dfrac{11}{2}}
$$

[Assistant]
Explanation:
Okay, so I need to solve the equation |x + 5| - |3x - 6| = 0. Hmm, absolute value equations can sometimes be tricky because they involve different cases depending on the sign of the expressions inside the absolute values. Let me think about how to approach this.

First, I remember that when dealing with equations involving absolute values, it's helpful to consider different intervals where the expressions inside the absolute values don't change sign. That means I need to find the critical points where each absolute value expression equals zero. For |x + 5|, the critical point is when x + 5 = 0, which is x = -5. For |3x - 6|, the critical point is when 3x - 6 = 0, which simplifies to x = 2. So these two critical points divide the number line into three intervals:

1. x < -5
2. -5 ≤ x < 2
3. x ≥ 2

I need to analyze each interval separately because the expressions inside the absolute values will have consistent signs in each interval, allowing me to remove the absolute value signs appropriately.

Let me start with the first interval: x < -5.

In this interval, x + 5 is negative because x is less than -5, so |x + 5| = -(x + 5). Similarly, 3x - 6: since x is less than -5, multiplying by 3 gives 3x < -15, so 3x - 6 < -15 - 6 = -21, which is definitely negative. Therefore, |3x - 6| = -(3x - 6) = -3x + 6.

So substituting into the original equation:

| x + 5 | - | 3x - 6 | = 0 becomes:

-(x + 5) - (-3x + 6) = 0

Let me simplify that:

- x - 5 + 3x - 6 = 0

Combine like terms:

(-x + 3x) + (-5 - 6) = 0

2x - 11 = 0

Solving for x:

2x = 11 => x = 11/2 = 5.5

Wait, but hold on! This solution x = 5.5 is supposed to be in the interval x < -5. But 5.5 is not less than -5. That means there's no solution in this interval. So, the first interval doesn't give us any valid solutions.

Okay, moving on to the second interval: -5 ≤ x < 2.

In this interval, x + 5 is non-negative because x is greater than or equal to -5, so |x + 5| = x + 5. Now, for 3x - 6: since x is less than 2, 3x is less than 6, so 3x - 6 is negative. Therefore, |3x - 6| = -(3x - 6) = -3x + 6.

Substituting into the original equation:

(x + 5) - (-3x + 6) = 0

Simplify:

x + 5 + 3x - 6 = 0

Combine like terms:

( x + 3x ) + (5 - 6) = 0

4x - 1 = 0

Solving for x:

4x = 1 => x = 1/4

Now, check if this solution is within the interval -5 ≤ x < 2. Since 1/4 is 0.25, which is between -5 and 2, this is a valid solution. So x = 1/4 is a solution in this interval.

Now, moving on to the third interval: x ≥ 2.

In this interval, x + 5 is positive because x is at least 2, so |x + 5| = x + 5. For 3x - 6: since x is greater than or equal to 2, 3x is at least 6, so 3x - 6 is non-negative. Therefore, |3x - 6| = 3x - 6.

Substituting into the original equation:

(x + 5) - (3x - 6) = 0

Simplify:

x + 5 - 3x + 6 = 0

Combine like terms:

( x - 3x ) + (5 + 6) = 0

-2x + 11 = 0

Solving for x:

-2x = -11 => x = (-11)/(-2) = 11/2 = 5.5

Now, check if this solution is within the interval x ≥ 2. Since 11/2 is 5.5, which is greater than or equal to 2, this is a valid solution. So x = 11/2 is another solution.

So, from the three intervals, we have two solutions: x = 1/4 and x = 11/2. The question asks for the largest possible value of x, so between 1/4 and 11/2, the larger one is 11/2.

But let me double-check my work to make sure I didn't make any mistakes. Let me verify both solutions in the original equation.

First, check x = 1/4:

Compute |x + 5| - |3x - 6|:

x + 5 = 1/4 + 5 = 1/4 + 20/4 = 21/4. Absolute value is 21/4.

3x - 6 = 3*(1/4) - 6 = 3/4 - 24/4 = -21/4. Absolute value is 21/4.

So, 21/4 - 21/4 = 0. Correct. So that works.

Now check x = 11/2:

x + 5 = 11/2 + 5 = 11/2 + 10/2 = 21/2. Absolute value is 21/2.

3x - 6 = 3*(11/2) - 6 = 33/2 - 12/2 = 21/2. Absolute value is 21/2.

So, 21/2 - 21/2 = 0. Correct. That also works.

So both solutions are valid. Therefore, the largest possible value of x is 11/2, which is 5.5. Since the question asks for an improper fraction, 11/2 is already an improper fraction.

Wait, but let me just make sure there are no other solutions. The critical points were at x = -5 and x = 2, so we covered all intervals. In each interval, we found solutions where applicable. So, yes, only two solutions. Therefore, 11/2 is the largest.

Alternatively, maybe there's another way to approach this equation? Let me think. The original equation is |x + 5| = |3x - 6|. So, maybe I can square both sides to eliminate the absolute values? Although squaring can sometimes introduce extraneous solutions, but maybe in this case it's manageable.

Let me try that approach. If |A| = |B|, then A^2 = B^2. So:

(x + 5)^2 = (3x - 6)^2

Expanding both sides:

Left side: x^2 + 10x + 25

Right side: 9x^2 - 36x + 36

Bring all terms to one side:

x^2 + 10x + 25 - 9x^2 + 36x - 36 = 0

Combine like terms:

(1x^2 - 9x^2) + (10x + 36x) + (25 - 36) = 0

-8x^2 + 46x - 11 = 0

Multiply both sides by -1 to make the leading coefficient positive:

8x^2 - 46x + 11 = 0

Now, solving this quadratic equation. Let me use the quadratic formula:

x = [46 ± sqrt( (-46)^2 - 4*8*11 )]/(2*8)

Calculate discriminant:

D = 2116 - 352 = 1764

sqrt(1764) = 42

Therefore:

x = [46 ± 42]/16

So two solutions:

First solution: (46 + 42)/16 = 88/16 = 11/2 (since 88 divided by 16 is 5.5, which is 11/2)

Second solution: (46 - 42)/16 = 4/16 = 1/4

So same solutions as before: 11/2 and 1/4. Therefore, confirming that the solutions are correct. Therefore, the largest value is 11/2.

Therefore, the answer is 11/2.

But just to make sure, let me check if there's any other way someone might approach this problem and maybe get confused. For example, someone might try to solve |x + 5| = |3x - 6| by considering cases where the expressions inside are positive or negative. But that's essentially what I did with the intervals. Alternatively, someone might think of the absolute values as distances, but I think that might complicate things more.

Alternatively, maybe graphing both |x + 5| and |3x - 6| and finding their intersection points? But since I already have algebraic solutions, and verified them, I think that's sufficient.

Another thing to check: when I squared both sides, sometimes you can introduce extraneous solutions, but in this case, both solutions worked when I checked them in the original equation, so they are valid. Therefore, no issues there.

So, conclusion: the largest possible value of x is 11/2.

**Final Answer**
The largest possible value of $ x $ is \boxed{\dfrac{11}{2}}.

To solve the equation:

$$
|x + 5| - |3x - 6| = 0
$$

we begin by recognizing that this is an absolute value equation. The key to solving such equations is to consider the **critical points** where the expressions inside the absolute values change sign. These points occur when:

- $ x + 5 = 0 \Rightarrow x = -5 $
- $ 3x - 6 = 0 \Rightarrow x = 2 $

These two points divide the number line into three intervals:

1. $ x < -5 $
2. $ -5 \leq x < 2 $
3. $ x \geq 2 $

We will analyze the equation in each interval, removing the absolute value signs appropriately based on the sign of the expressions.

---

### **Interval 1: $ x < -5 $**

In this interval:
- $ x + 5 < 0 \Rightarrow |x + 5| = -(x + 5) $
- $ 3x - 6 < 0 \Rightarrow |3x - 6| = -(3x - 6) $

Substitute into the equation:

$$
-(x + 5) - (-(3x - 6)) = 0 \Rightarrow -x - 5 + 3x - 6 = 0
$$

Simplify:

$$
2x - 11 = 0 \Rightarrow x = \frac{11}{2}
$$

But $ \frac{11}{2} = 5.5 $, which is **not** in the interval $ x < -5 $. So, **no solution** in this interval.

---

### **Interval 2: $ -5 \leq x < 2 $**

In this interval:
- $ x + 5 \geq 0 \Rightarrow |x + 5| = x + 5 $
- $ 3x - 6 < 0 \Rightarrow |3x - 6| = -(3x - 6) $

Substitute into the equation:

$$
(x + 5) - (-(3x - 6)) = 0 \Rightarrow x + 5 + 3x - 6 = 0
$$

Simplify:

$$
4x - 1 = 0 \Rightarrow x = \frac{1}{4}
$$

Check: $ \frac{1}{4} $ is in the interval $ -5 \leq x < 2 $, so this is a **valid solution**.

---

### **Interval 3: $ x \geq 2 $**

In this interval:
- $ x + 5 \geq 0 \Rightarrow |x + 5| = x + 5 $
- $ 3x - 6 \geq 0 \Rightarrow |3x - 6| = 3x - 6 $

Substitute into the equation:

$$
(x + 5) - (3x - 6) = 0 \Rightarrow x + 5 - 3x + 6 = 0
$$

Simplify:

$$
-2x + 11 = 0 \Rightarrow x = \frac{11}{2}
$$

Check: $ \frac{11}{2} = 5.5 $, which is in the interval $ x \geq 2 $, so this is also a **valid solution**.

---

### **Final Step: Determine the Largest Value**

From the valid solutions:

- $ x = \frac{1}{4} $
- $ x = \frac{11}{2} $

The **largest** value is:

$$
\boxed{\dfrac{11}{2}}
$$

---

### **Verification (Optional but Recommended)**

Let’s verify both solutions in the original equation:

- For $ x = \frac{1}{4} $:
  $$
  |x + 5| = \left| \frac{1}{4} + 5 \right| = \left| \frac{21}{4} \right| = \frac{21}{4}
  $$
  $$
  |3x - 6| = \left| \frac{3}{4} - 6 \right| = \left| -\frac{21}{4} \right| = \frac{21}{4}
  $$
  $$
  \frac{21}{4} - \frac{21}{4} = 0 \quad \text{✓}
  $$

- For $ x = \frac{11}{2} $:
  $$
  |x + 5| = \left| \frac{11}{2} + 5 \right| = \left| \frac{21}{2} \right| = \frac{21}{2}
  $$
  $$
  |3x - 6| = \left| \frac{33}{2} - 6 \right| = \left| \frac{21}{2} \right| = \frac{21}{2}
  $$
  $$
  \frac{21}{2} - \frac{21}{2} = 0 \quad \text{✓}
  $$

Both solutions are valid, and the **largest** is:

$$
\boxed{\dfrac{11}{2}}
$$

Solution:
We begin by moving the second inequality to the right side of the equation, giving us $|x+5|=|3x-6|$. From here, we can split the equation into two separate cases.  For the first case, note that if $x+5$ and $3x-6$ have the same sign, then $x+5=3x-6$:

Case 1: \begin{align*} x+5&=3x-6
\\\Rightarrow \qquad -2x&=-11
\\\Rightarrow \qquad x&=\frac{11}{2}
\end{align*}If we plug this value of $x$ back into the original equation to check our answer, we get that $\left|\frac{11}{2}+5\right|-\left|3\left(\frac{11}{2}\right)-6\right|=0$ or $0=0$. Since this is true, we can accept $x=\frac{11}{2}$ as a valid solution.

For case two, note that if $x+5$ has a different sign than $3x-6$, then $x+5=-(3x-6)$.

Case 2: \begin{align*} x+5&=-(3x-6)
\\ x+5&=-3x+6
\\\Rightarrow \qquad 4x&=1
\\\Rightarrow \qquad x&=\frac{1}{4}
\end{align*}If we plug this value of $x$ back into the original equation to check our answer, we get that $\left|\frac{1}{4}+5\right|-\left|3\left(\frac{1}{4}\right)-6\right|=0$, which also gives us $0=0$. This is always true, so we can accept $x=\frac{1}{4}$ as a valid solution as well. Thus, our two possible solutions are $\frac{1}{4}$ and $\frac{11}{2}$. Since the question asks for the largest possible value of $x$, our final solution is $\boxed{\frac{11}{2}}$.