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 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).

So substituting into the original equation:

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

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 contribute any 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. For 3x - 6, since x is less than 2, 3x is less than 6, so 3x - 6 is 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

4x - 1 = 0

Solving for x:

4x = 1 => x = 1/4

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

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 at least 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. 11/2 is 5.5, which is indeed greater than or equal to 2. So, this is also a valid solution. Therefore, 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 is 11/2, which is 5.5. Since the question asks for an improper fraction, 11/2 is already an improper fraction. So, the answer should be 11/2.

But let me just think again if there's another way to approach this problem, maybe by rearranging the equation first?

Original equation: |x + 5| - |3x - 6| = 0. Which can be rewritten as |x + 5| = |3x - 6|.

So, another approach is to solve |x + 5| = |3x - 6|. This equation implies that either x + 5 = 3x - 6 or x + 5 = -(3x - 6). Let me try this method and see if I get the same solutions.

First case: x + 5 = 3x - 6.

Solving:

x + 5 = 3x - 6

Subtract x from both sides:

5 = 2x - 6

Add 6 to both sides:

11 = 2x => x = 11/2. Which is the same as before.

Second case: x + 5 = - (3x - 6) => x + 5 = -3x + 6.

Solving:

x + 5 = -3x + 6

Add 3x to both sides:

4x + 5 = 6

Subtract 5:

4x = 1 => x = 1/4. Which is the same as before.

So, same solutions. Therefore, confirming that the solutions are x = 1/4 and x = 11/2, so the largest is 11/2. Therefore, the answer is 11/2.

But just to be thorough, let me check if there are any other cases or if I missed something. For example, sometimes when dealing with absolute values, there can be more cases, but in this case, since we have two absolute values, the standard approach is to consider the critical points and split into intervals. Since we did that and also solved by squaring both sides (though squaring might not be necessary here), but since we have |a| = |b|, which implies a = b or a = -b, which is exactly what I did in the second approach. So, both methods lead to the same solutions.

Alternatively, another way to think about |x + 5| = |3x - 6| is that the distance from x to -5 is equal to the distance from x to 2 (since |x - (-5)| = |x - 2|). Wait, hold on, no. Wait, |x + 5| is the distance from x to -5, and |3x - 6| is the distance from 3x to 6, but that's not the same as distance from x to something. Wait, maybe not. Let me think.

Wait, |3x - 6| can be written as 3|x - 2|. Because |3x - 6| = |3(x - 2)| = 3|x - 2|. So, the original equation |x + 5| = 3|x - 2|. So, maybe that's another way to think about it. So, |x + 5| = 3|x - 2|. But I don't know if that helps directly, but maybe.

But perhaps not necessary. Since we already have the solutions, and verified them, maybe that's sufficient.

Alternatively, if I graph both |x + 5| and |3x - 6|, their intersection points would be the solutions. But since I can't graph here, but I can think about the behavior.

But again, since we have already solved it through two methods and verified the solutions, I think it's safe to say that 11/2 is the correct answer.

Therefore, 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:

$$
-(x + 5) - [-(3x - 6)] = 0 \\
- x - 5 + 3x - 6 = 0 \\
2x - 11 = 0 \Rightarrow x = \frac{11}{2}
$$

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:

$$
(x + 5) - [-(3x - 6)] = 0 \\
x + 5 + 3x - 6 = 0 \\
4x - 1 = 0 \Rightarrow x = \frac{1}{4}
$$

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:

$$
(x + 5) - (3x - 6) = 0 \\
x + 5 - 3x + 6 = 0 \\
-2x + 11 = 0 \Rightarrow x = \frac{11}{2}
$$

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

---

### **Conclusion**

We found two valid solutions:

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

The **largest possible value** of $ x $ 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 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).

So substituting into the original equation:

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

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 contribute any 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. For 3x - 6, since x is less than 2, 3x is less than 6, so 3x - 6 is 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

4x - 1 = 0

Solving for x:

4x = 1 => x = 1/4

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

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 at least 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. 11/2 is 5.5, which is indeed greater than or equal to 2. So, this is also a valid solution. Therefore, 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 is 11/2, which is 5.5. Since the question asks for an improper fraction, 11/2 is already an improper fraction. So, the answer should be 11/2.

But let me just think again if there's another way to approach this problem, maybe by rearranging the equation first?

Original equation: |x + 5| - |3x - 6| = 0. Which can be rewritten as |x + 5| = |3x - 6|.

So, another approach is to solve |x + 5| = |3x - 6|. This equation implies that either x + 5 = 3x - 6 or x + 5 = -(3x - 6). Let me try this method and see if I get the same solutions.

First case: x + 5 = 3x - 6.

Solving:

x + 5 = 3x - 6

Subtract x from both sides:

5 = 2x - 6

Add 6 to both sides:

11 = 2x => x = 11/2. Which is the same as before.

Second case: x + 5 = - (3x - 6) => x + 5 = -3x + 6.

Solving:

x + 5 = -3x + 6

Add 3x to both sides:

4x + 5 = 6

Subtract 5:

4x = 1 => x = 1/4. Which is the same as before.

So, same solutions. Therefore, confirming that the solutions are x = 1/4 and x = 11/2, so the largest is 11/2. Therefore, the answer is 11/2.

But just to be thorough, let me check if there are any other cases or if I missed something. For example, sometimes when dealing with absolute values, there can be more cases, but in this case, since we have two absolute values, the standard approach is to consider the critical points and split into intervals. Since we did that and also solved by squaring both sides (though squaring might not be necessary here), but since we have |a| = |b|, which implies a = b or a = -b, which is exactly what I did in the second approach. So, both methods lead to the same solutions.

Alternatively, another way to think about |x + 5| = |3x - 6| is that the distance from x to -5 is equal to the distance from x to 2 (since |x - (-5)| = |x - 2|). Wait, hold on, no. Wait, |x + 5| is the distance from x to -5, and |3x - 6| is the distance from 3x to 6, but that's not the same as distance from x to something. Wait, maybe not. Let me think.

Wait, |3x - 6| can be written as 3|x - 2|. Because |3x - 6| = |3(x - 2)| = 3|x - 2|. So, the original equation |x + 5| = 3|x - 2|. So, maybe that's another way to think about it. So, |x + 5| = 3|x - 2|. But I don't know if that helps directly, but maybe.

But perhaps not necessary. Since we already have the solutions, and verified them, maybe that's sufficient.

Alternatively, if I graph both |x + 5| and |3x - 6|, their intersection points would be the solutions. But since I can't graph here, but I can think about the behavior.

But again, since we have already solved it through two methods and verified the solutions, I think it's safe to say that 11/2 is the correct answer.

Therefore, 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:

$$
-(x + 5) - [-(3x - 6)] = 0 \\
- x - 5 + 3x - 6 = 0 \\
2x - 11 = 0 \Rightarrow x = \frac{11}{2}
$$

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:

$$
(x + 5) - [-(3x - 6)] = 0 \\
x + 5 + 3x - 6 = 0 \\
4x - 1 = 0 \Rightarrow x = \frac{1}{4}
$$

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:

$$
(x + 5) - (3x - 6) = 0 \\
x + 5 - 3x + 6 = 0 \\
-2x + 11 = 0 \Rightarrow x = \frac{11}{2}
$$

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

---

### **Conclusion**

We found two valid solutions:

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

The **largest possible value** of $ x $ 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 $\boxed{\frac{11}{2}}$.


Explanation:
Okay, so I need to evaluate this expression: the ceiling of (3.6) squared minus the ceiling of 3.6 squared. Let me write that down to make sure I have it right. It's ⌈(3.6)²⌉ - (⌈3.6⌉)². Alright, let me break this down step by step.

First, let me recall what the ceiling function does. The ceiling of a number is the smallest integer greater than or equal to that number. So, for example, the ceiling of 3.2 would be 4, because 4 is the next integer higher than 3.2. Similarly, the ceiling of 3.9 would also be 4, and the ceiling of 4 itself is 4. Got it.

So, the expression has two parts: the first part is the ceiling of (3.6) squared, and the second part is the ceiling of 3.6 squared. Then we subtract the second part from the first. Let me compute each part separately.

Starting with the first part: ⌈(3.6)²⌉. Let me calculate (3.6) squared first. 3.6 times 3.6. Hmm, 3 times 3 is 9, 3 times 0.6 is 1.8, 0.6 times 3 is another 1.8, and 0.6 times 0.6 is 0.36. Adding those up: 9 + 1.8 + 1.8 + 0.36. Let me compute that. 9 + 1.8 is 10.8, plus another 1.8 is 12.6, plus 0.36 is 12.96. So, (3.6)² equals 12.96. Now, taking the ceiling of that. Since 12.96 is between 12 and 13, the ceiling would be 13. So, the first part is 13.

Now, moving on to the second part: (⌈3.6⌉)². Let me compute the ceiling of 3.6 first. As I thought earlier, the ceiling of 3.6 is 4 because it's the smallest integer greater than or equal to 3.6. Then, squaring that result: 4 squared is 16. So, the second part is 16.

Now, subtracting the second part from the first part: 13 - 16. That would be... let me do the subtraction. 13 minus 16 is -3. Wait, so the result is -3? Let me check my steps again to make sure I didn't make a mistake.

First, calculating (3.6)^2. Let me verify that again. 3.6 times 3.6. Alternatively, I can think of 3.6 as 36/10. So, (36/10)^2 is 1296/100, which is 12.96. Yep, that's correct. Then ceiling of 12.96 is indeed 13. Then ceiling of 3.6 is 4, squared is 16. Then 13 - 16 is -3. That seems straightforward.

Wait, but let me make sure I didn't mix up the order of operations or something. The original expression is ⌈(3.6)^2⌉ - (⌈3.6⌉)^2. So, yes, first compute (3.6)^2, take ceiling, then compute ceiling of 3.6, square it, then subtract the second result from the first. So, 13 - 16 = -3. That seems right.

But just to be thorough, let me check if there's any alternative interpretation. For example, could someone misread the expression as ceiling of (3.6 squared minus ceiling of 3.6 squared)? But no, the parentheses make it clear. The first term is ceiling of (3.6 squared), and the second term is (ceiling of 3.6) squared. So, the way it's written, it's definitely ⌈(3.6)^2⌉ - (⌈3.6⌉)^2.

Alternatively, maybe I miscalculated the ceiling of 3.6? Let me check that again. 3.6 is between 3 and 4. The ceiling function rounds up to the next integer, so yes, ceiling of 3.6 is 4. So, squaring that gives 16. Then ceiling of 12.96 is 13. So, 13 - 16 is indeed -3.

Wait, but maybe I made a mistake in calculating (3.6)^2? Let me do that again. 3.6 times 3.6. Let me compute 3 * 3.6 first. 3 * 3.6 is 10.8. Then 0.6 * 3.6 is... 0.6 * 3 is 1.8, and 0.6 * 0.6 is 0.36, so total is 1.8 + 0.36 = 2.16. Then adding 10.8 + 2.16 gives 12.96. Yep, same result. So that's correct.

Alternatively, maybe I can use decimal multiplication. 3.6 * 3.6. Multiply 36 * 36 first, which is 1296, and since each 3.6 has one decimal place, total two decimal places, so 12.96. Yep, same thing. So that's correct.

So, ceiling of 12.96 is 13. Then ceiling of 3.6 is 4, squared is 16. 13 - 16 is -3. So, the answer is -3. Is there any chance that the ceiling function could be different? For example, if the number was exactly an integer, but 3.6 isn't. So, ceiling of 3.6 is definitely 4. And ceiling of 12.96 is 13. So, I think that's right.

Alternatively, maybe the problem is testing something else? Let me think. For example, sometimes with ceiling functions, people might confuse with floor functions, but no, the problem specifically says ceiling. So, I think my reasoning is correct.

Alternatively, maybe I need to check if there's a different order of operations? Let me check the original problem again: Evaluate ⌈(3.6)^2⌉ - (⌈3.6⌉)^2. So, the first term is ceiling of (3.6 squared), the second term is (ceiling of 3.6) squared. So, the operations are as I did. So, 13 - 16 = -3.

Alternatively, maybe the problem is written differently? For example, if it was ⌈(3.6)^2 - (⌈3.6⌉)^2⌉, but no, the original problem is written as ⌈(3.6)^2⌉ - (⌈3.6⌉)^2. So, parentheses are around the 3.6 squared in the first term, and the second term is ceiling of 3.6 squared. So, no, the way it's written, it's definitely the difference between those two terms.

Therefore, I think my answer is correct. The result is -3. Let me just write that in the box as requested.

**Final Answer**
\boxed{-3}

To evaluate the expression:

$$
\lceil (3.6)^2 \rceil - ( \lceil 3.6 \rceil )^2
$$

we proceed step by step, carefully applying the ceiling function and arithmetic operations.

---

### Step 1: Compute $ (3.6)^2 $

We calculate the square of 3.6:

$$
(3.6)^2 = 3.6 \times 3.6 = 12.96
$$

---

### Step 2: Apply the ceiling function to $ (3.6)^2 $

The ceiling function $ \lceil x \rceil $ gives the smallest integer greater than or equal to $ x $. Since $ 12.96 $ is not an integer, we round it up:

$$
\lceil 12.96 \rceil = 13
$$

---

### Step 3: Compute $ \lceil 3.6 \rceil $

Again, applying the ceiling function to 3.6:

$$
\lceil 3.6 \rceil = 4
$$

---

### Step 4: Square the result from Step 3

$$
( \lceil 3.6 \rceil )^2 = 4^2 = 16
$$

---

### Step 5: Subtract the results

Now, subtract the second result from the first:

$$
\lceil (3.6)^2 \rceil - ( \lceil 3.6 \rceil )^2 = 13 - 16 = -3
$$

---

### Final Answer

$$
\boxed{-3}
$$

[Assistant]
Explanation:
Okay, so I need to evaluate this expression: the ceiling of (3.6) squared minus the ceiling of 3.6 squared. Let me write that down to make sure I have it right. It's ⌈(3.6)²⌉ - (⌈3.6⌉)². Alright, let me break this down step by step.

First, let me recall what the ceiling function does. The ceiling of a number is the smallest integer greater than or equal to that number. So, for example, the ceiling of 3.2 would be 4, because 4 is the next integer higher than 3.2. Similarly, the ceiling of 3.9 would also be 4, and the ceiling of 4 itself is 4. Got it.

So, the expression has two parts: the first part is the ceiling of (3.6) squared, and the second part is the ceiling of 3.6 squared. Then we subtract the second part from the first. Let me compute each part separately.

Starting with the first part: ⌈(3.6)²⌉. Let me calculate (3.6) squared first. 3.6 times 3.6. Hmm, 3 times 3 is 9, 3 times 0.6 is 1.8, 0.6 times 3 is another 1.8, and 0.6 times 0.6 is 0.36. Adding those up: 9 + 1.8 + 1.8 + 0.36. Let me compute that. 9 + 1.8 is 10.8, plus another 1.8 is 12.6, plus 0.36 is 12.96. So, (3.6)² equals 12.96. Now, taking the ceiling of that. Since 12.96 is between 12 and 13, the ceiling would be 13. So, the first part is 13.

Now, moving on to the second part: (⌈3.6⌉)². Let me compute the ceiling of 3.6 first. As I thought earlier, the ceiling of 3.6 is 4 because it's the smallest integer greater than or equal to 3.6. Then, squaring that result: 4 squared is 16. So, the second part is 16.

Now, subtracting the second part from the first part: 13 - 16. That would be... let me do the subtraction. 13 minus 16 is -3. Wait, so the result is -3? Let me check my steps again to make sure I didn't make a mistake.

First, calculating (3.6)^2. Let me verify that again. 3.6 times 3.6. Alternatively, I can think of 3.6 as 36/10. So, (36/10)^2 is 1296/100, which is 12.96. Yep, that's correct. Then ceiling of 12.96 is indeed 13. Then ceiling of 3.6 is 4, squared is 16. Then 13 - 16 is -3. That seems straightforward.

Wait, but let me make sure I didn't mix up the order of operations or something. The original expression is ⌈(3.6)^2⌉ - (⌈3.6⌉)^2. So, yes, first compute (3.6)^2, take ceiling, then compute ceiling of 3.6, square it, then subtract the second result from the first. So, 13 - 16 = -3. That seems right.

But just to be thorough, let me check if there's any alternative interpretation. For example, could someone misread the expression as ceiling of (3.6 squared minus ceiling of 3.6 squared)? But no, the parentheses make it clear. The first term is ceiling of (3.6 squared), and the second term is (ceiling of 3.6) squared. So, the way it's written, it's definitely ⌈(3.6)^2⌉ - (⌈3.6⌉)^2.

Alternatively, maybe I miscalculated the ceiling of 3.6? Let me check that again. 3.6 is between 3 and 4. The ceiling function rounds up to the next integer, so yes, ceiling of 3.6 is 4. So, squaring that gives 16. Then ceiling of 12.96 is 13. So, 13 - 16 is indeed -3.

Wait, but maybe I made a mistake in calculating (3.6)^2? Let me do that again. 3.6 times 3.6. Let me compute 3 * 3.6 first. 3 * 3.6 is 10.8. Then 0.6 * 3.6 is... 0.6 * 3 is 1.8, and 0.6 * 0.6 is 0.36, so total is 1.8 + 0.36 = 2.16. Then adding 10.8 + 2.16 gives 12.96. Yep, same result. So that's correct.

Alternatively, maybe I can use decimal multiplication. 3.6 * 3.6. Multiply 36 * 36 first, which is 1296, and since each 3.6 has one decimal place, total two decimal places, so 12.96. Yep, same thing. So that's correct.

So, ceiling of 12.96 is 13. Then ceiling of 3.6 is 4, squared is 16. 13 - 16 is -3. So, the answer is -3. Is there any chance that the ceiling function could be different? For example, if the number was exactly an integer, but 3.6 isn't. So, ceiling of 3.6 is definitely 4. And ceiling of 12.96 is 13. So, I think that's right.

Alternatively, maybe the problem is testing something else? Let me think. For example, sometimes with ceiling functions, people might confuse with floor functions, but no, the problem specifically says ceiling. So, I think my reasoning is correct.

Alternatively, maybe I need to check if there's a different order of operations? Let me check the original problem again: Evaluate ⌈(3.6)^2⌉ - (⌈3.6⌉)^2. So, the first term is ceiling of (3.6 squared), the second term is (ceiling of 3.6) squared. So, the operations are as I did. So, 13 - 16 = -3.

Alternatively, maybe the problem is written differently? For example, if it was ⌈(3.6)^2 - (⌈3.6⌉)^2⌉, but no, the original problem is written as ⌈(3.6)^2⌉ - (⌈3.6⌉)^2. So, parentheses are around the 3.6 squared in the first term, and the second term is ceiling of 3.6 squared. So, no, the way it's written, it's definitely the difference between those two terms.

Therefore, I think my answer is correct. The result is -3. Let me just write that in the box as requested.

**Final Answer**
\boxed{-3}

To evaluate the expression:

$$
\lceil (3.6)^2 \rceil - ( \lceil 3.6 \rceil )^2
$$

we proceed step by step, carefully applying the ceiling function and arithmetic operations.

---

### Step 1: Compute $ (3.6)^2 $

We calculate the square of 3.6:

$$
(3.6)^2 = 3.6 \times 3.6 = 12.96
$$

---

### Step 2: Apply the ceiling function to $ (3.6)^2 $

The ceiling function $ \lceil x \rceil $ gives the smallest integer greater than or equal to $ x $. Since $ 12.96 $ is not an integer, we round it up:

$$
\lceil 12.96 \rceil = 13
$$

---

### Step 3: Compute $ \lceil 3.6 \rceil $

Again, applying the ceiling function to 3.6:

$$
\lceil 3.6 \rceil = 4
$$

---

### Step 4: Square the result from Step 3

$$
( \lceil 3.6 \rceil )^2 = 4^2 = 16
$$

---

### Step 5: Subtract the results

Now, subtract the second result from the first:

$$
\lceil (3.6)^2 \rceil - ( \lceil 3.6 \rceil )^2 = 13 - 16 = -3
$$

---

### Final Answer

$$
\boxed{-3}
$$

Solution: $\lceil (3.6)^2 \rceil = \lceil 12.96 \rceil = 13$ because the least integer greater than $12.96$ is $13$. $( \lceil 3.6 \rceil ) ^2 = 4^2 = 16$ because the least integer greater than $3.6$ is $4$. Therefore, the answer is $13-16=\boxed{-3}$.