Explanation:
Okay, so I need to find an integer x that satisfies the inequality 1/4 < x/7 < 1/3. Hmm, let me think about how to approach this.

First, maybe I should try to solve the inequality for x. Let me write down the inequality again:

1/4 < x/7 < 1/3.

I need to find the value of x that makes this true. Since all the terms are positive, I can multiply all parts of the inequality by 7 to eliminate the denominator. Let me do that step by step.

Starting with the left part of the inequality: 1/4 < x/7. If I multiply both sides by 7, that would give me (1/4)*7 < x. Let me calculate that. 7 divided by 4 is 1.75. So, 1.75 < x.

Now, the right part of the inequality: x/7 < 1/3. Multiplying both sides by 7 gives x < (1/3)*7. Calculating that, 7 divided by 3 is approximately 2.333... So, x < 2.333...

Putting both parts together, we have 1.75 < x < 2.333...

But x has to be an integer. Let me think. The integers between 1.75 and 2.333... Well, 1.75 is between 1 and 2, and 2.333... is between 2 and 3. So the only integer that lies strictly between 1.75 and 2.333 is 2. Because 2 is greater than 1.75 and less than 2.333...

Wait, let me check if that's correct. Let me verify by plugging x = 2 into the original inequality.

Original inequality: 1/4 < 2/7 < 1/3.

Calculating 1/4 is 0.25, 2/7 is approximately 0.2857, and 1/3 is approximately 0.3333. So 0.25 < 0.2857 < 0.3333. Yes, that's true. So x = 2 satisfies the inequality.

But let me make sure there are no other integers in that interval. The next integer above 2 is 3. Let me check x = 3. Then 3/7 is approximately 0.4286, which is greater than 1/3 (0.3333), so that doesn't work. The integer below 2 is 1. Let me check x = 1. Then 1/7 is approximately 0.1429, which is less than 1/4 (0.25), so that doesn't work either. Therefore, the only integer that satisfies the inequality is 2.

Wait, but let me think again. Maybe I should approach this by cross-multiplying or something else to confirm?

Alternatively, maybe I can convert the fractions to have the same denominator or compare them differently. Let me try that.

Original inequality: 1/4 < x/7 < 1/3.

Let me first compare 1/4 and 1/3. Since 1/4 is 0.25 and 1/3 is approximately 0.333..., so the value of x/7 has to be between these two. So, if I can find x such that when divided by 7, it's between 0.25 and 0.333...

Alternatively, maybe I can write the inequalities as:

For the left inequality: 1/4 < x/7. Multiply both sides by 7: 7/4 < x. Which is 1.75 < x.

For the right inequality: x/7 < 1/3. Multiply both sides by 7: x < 7/3. Which is approximately 2.333...

So again, x must be between 1.75 and 2.333..., so the only integer is 2. So that seems consistent.

Alternatively, maybe I can think of fractions. Let me think. If I have 1/4 < x/7 < 1/3, then cross-multiplying (since all numbers are positive, inequalities preserve direction).

For the left inequality: 1/4 < x/7. Cross-multiplying gives 7*1 < 4x => 7 < 4x => x > 7/4 = 1.75.

For the right inequality: x/7 < 1/3. Cross-multiplying gives 3x < 7 => x < 7/3 ≈ 2.333...

So again, same result. Therefore, x must be greater than 1.75 and less than 2.333..., so integer x is 2.

Alternatively, maybe I can think of fractions with denominator 7. Let me see. Let me think of fractions between 1/4 and 1/3 with denominator 7. So, 1/4 is equal to 7/28, and 1/3 is equal to 7/21? Wait, no. Wait, 1/4 is 7/28? Wait, 1/4 is equal to 7/28? Wait, 1/4 is 7/28? Wait, 1/4 is 7/28? Let me check. 1/4 is equal to 7/28? No, 1/4 is equal to 7/28? Wait, 1/4 is equal to 7/28? Wait, 1/4 is equal to 7/28? Let me compute 7 divided by 28. That is 0.25, which is 1/4. Yes, so 1/4 is 7/28. Similarly, 1/3 is equal to 7/21? Wait, 7 divided by 21 is 1/3. Yes. So, the inequality is 7/28 < x/7 < 7/21.

But maybe this isn't helpful. Alternatively, maybe I can write x/7 as a fraction with denominator 7. So, x/7 is just x over 7. So, if I want to find x such that 1/4 < x/7 < 1/3, then maybe I can find x such that when you divide by 7, it's between those two fractions.

Alternatively, maybe I can find a common denominator for 1/4 and 1/3. The least common denominator is 12. So, 1/4 is 3/12 and 1/3 is 4/12. Therefore, the inequality is 3/12 < x/7 < 4/12. But I don't know if that helps directly. Maybe I can cross-multiply?

Alternatively, maybe I can think of x as being between 7*(1/4) and 7*(1/3). Which is exactly what I did earlier. So, 7*(1/4) is 7/4 = 1.75 and 7*(1/3) is 7/3 ≈ 2.333...

So, again, x must be between 1.75 and 2.333..., so integer x is 2.

Alternatively, maybe I can think of fractions between 1/4 and 1/3. Let me think. For example, 1/4 is 0.25, 1/3 is approximately 0.333. So, what fractions between these two? Well, 2/7 is approximately 0.2857, which is between them. Is there another fraction? For example, 3/10 is 0.3, which is also between them. But the question is about x/7, so x must be such that when divided by 7, it's between 1/4 and 1/3. Therefore, x must be such that x/7 is equal to some fraction between those two. But since x is an integer, the only possible x is 2, because 2/7 is approximately 0.2857, which is between 0.25 and 0.333.

Alternatively, maybe I can check if there are other integers. For example, if x is 2, then 2/7 is between 1/4 and 1/3. If x is 1, then 1/7 is less than 1/4. If x is 3, then 3/7 is approximately 0.428, which is greater than 1/3. So, no other integers.

Wait, but let me check if there's another way. Maybe by converting the inequalities to decimals?

Original inequality: 1/4 is 0.25, 1/3 is approximately 0.3333. So, x/7 must be between 0.25 and 0.3333. Therefore, multiplying all parts by 7, we get x between 0.25*7 = 1.75 and 0.3333*7 ≈ 2.333. So again, x must be between 1.75 and 2.333. So, the only integer is 2.

Therefore, I think x is 2.

But let me check once more. Maybe I made a mistake in calculation? Let me verify:

If x = 2, then 2 divided by 7 is approximately 0.2857.

Now, 1/4 is 0.25, and 1/3 is approximately 0.3333. So 0.25 < 0.2857 < 0.3333. Yes, that's correct.

If x = 1, then 1/7 ≈ 0.1429, which is less than 0.25.

If x = 3, then 3/7 ≈ 0.4286, which is greater than 0.3333.

Therefore, there are no other integers.

Alternatively, maybe I can think of fractions with denominator 7. Let me think. The fractions with denominator 7 are 0/7, 1/7, 2/7, 3/7, etc. So, between 1/4 and 1/3, which is between 0.25 and 0.333..., so looking at the fractions with denominator 7:

1/7 ≈ 0.1429 (too small)

2/7 ≈ 0.2857 (between 0.25 and 0.333)

3/7 ≈ 0.4286 (too big)

So, only 2/7 is between 1/4 and 1/3. Therefore, x must be 2.

Therefore, the answer is 2.

I think that's thorough. I considered multiple approaches: solving the inequality algebraically, converting to decimals, checking fractions with denominator 7, and verifying by plugging in values. All methods lead to the conclusion that x = 2 is the only integer satisfying the inequality.

**Final Answer**
The integer $ x $ is \boxed{2}.

To solve the inequality:

$$
\frac{1}{4} < \frac{x}{7} < \frac{1}{3}
$$

we aim to find an integer $ x $ that satisfies this condition.

---

### Step 1: Eliminate the Denominator

To simplify the inequality, we can multiply all parts by 7 (since 7 is positive, the inequality direction remains unchanged):

$$
7 \cdot \frac{1}{4} < x < 7 \cdot \frac{1}{3}
$$

$$
\frac{7}{4} < x < \frac{7}{3}
$$

Now compute the decimal values:

$$
\frac{7}{4} = 1.75 \quad \text{and} \quad \frac{7}{3} \approx 2.333\ldots
$$

So the inequality becomes:

$$
1.75 < x < 2.333\ldots
$$

---

### Step 2: Identify the Integer in the Interval

We are looking for an **integer** $ x $ that lies strictly between 1.75 and 2.333...

The only integer in this interval is:

$$
x = 2
$$

---

### Step 3: Verify the Solution

Let’s plug $ x = 2 $ back into the original inequality to confirm:

$$
\frac{1}{4} < \frac{2}{7} < \frac{1}{3}
$$

Convert to decimals for comparison:

- $ \frac{1}{4} = 0.25 $
- $ \frac{2}{7} \approx 0.2857 $
- $ \frac{1}{3} \approx 0.3333 $

So:

$$
0.25 < 0.2857 < 0.3333
$$

This confirms that $ x = 2 $ satisfies the inequality.

---

### Step 4: Check for Other Possible Integers

- $ x = 1 $: $ \frac{1}{7} \approx 0.1429 $ → too small
- $ x = 3 $: $ \frac{3}{7} \approx 0.4286 $ → too large

Thus, **no other integers** satisfy the inequality.

---

### Final Answer

$$
\boxed{2}
$$

[Assistant]
Explanation:
Okay, so I need to find an integer x that satisfies the inequality 1/4 < x/7 < 1/3. Hmm, let me think about how to approach this.

First, maybe I should try to solve the inequality for x. Let me write down the inequality again:

1/4 < x/7 < 1/3.

I need to find the value of x that makes this true. Since all the terms are positive, I can multiply all parts of the inequality by 7 to eliminate the denominator. Let me do that step by step.

Starting with the left part of the inequality: 1/4 < x/7. If I multiply both sides by 7, that would give me (1/4)*7 < x. Let me calculate that. 7 divided by 4 is 1.75. So, 1.75 < x.

Now, the right part of the inequality: x/7 < 1/3. Multiplying both sides by 7 gives x < (1/3)*7. Calculating that, 7 divided by 3 is approximately 2.333... So, x < 2.333...

Putting both parts together, we have 1.75 < x < 2.333...

But x has to be an integer. Let me think. The integers between 1.75 and 2.333... Well, 1.75 is between 1 and 2, and 2.333... is between 2 and 3. So the only integer that lies strictly between 1.75 and 2.333 is 2. Because 2 is greater than 1.75 and less than 2.333...

Wait, let me check if that's correct. Let me verify by plugging x = 2 into the original inequality.

Original inequality: 1/4 < 2/7 < 1/3.

Calculating 1/4 is 0.25, 2/7 is approximately 0.2857, and 1/3 is approximately 0.3333. So 0.25 < 0.2857 < 0.3333. Yes, that's true. So x = 2 satisfies the inequality.

But let me make sure there are no other integers in that interval. The next integer above 2 is 3. Let me check x = 3. Then 3/7 is approximately 0.4286, which is greater than 1/3 (0.3333), so that doesn't work. The integer below 2 is 1. Let me check x = 1. Then 1/7 is approximately 0.1429, which is less than 1/4 (0.25), so that doesn't work either. Therefore, the only integer that satisfies the inequality is 2.

Wait, but let me think again. Maybe I should approach this by cross-multiplying or something else to confirm?

Alternatively, maybe I can convert the fractions to have the same denominator or compare them differently. Let me try that.

Original inequality: 1/4 < x/7 < 1/3.

Let me first compare 1/4 and 1/3. Since 1/4 is 0.25 and 1/3 is approximately 0.333..., so the value of x/7 has to be between these two. So, if I can find x such that when divided by 7, it's between 0.25 and 0.333...

Alternatively, maybe I can write the inequalities as:

For the left inequality: 1/4 < x/7. Multiply both sides by 7: 7/4 < x. Which is 1.75 < x.

For the right inequality: x/7 < 1/3. Multiply both sides by 7: x < 7/3. Which is approximately 2.333...

So again, x must be between 1.75 and 2.333..., so the only integer is 2. So that seems consistent.

Alternatively, maybe I can think of fractions. Let me think. If I have 1/4 < x/7 < 1/3, then cross-multiplying (since all numbers are positive, inequalities preserve direction).

For the left inequality: 1/4 < x/7. Cross-multiplying gives 7*1 < 4x => 7 < 4x => x > 7/4 = 1.75.

For the right inequality: x/7 < 1/3. Cross-multiplying gives 3x < 7 => x < 7/3 ≈ 2.333...

So again, same result. Therefore, x must be greater than 1.75 and less than 2.333..., so integer x is 2.

Alternatively, maybe I can think of fractions with denominator 7. Let me see. Let me think of fractions between 1/4 and 1/3 with denominator 7. So, 1/4 is equal to 7/28, and 1/3 is equal to 7/21? Wait, no. Wait, 1/4 is 7/28? Wait, 1/4 is equal to 7/28? Wait, 1/4 is 7/28? Wait, 1/4 is 7/28? Let me check. 1/4 is equal to 7/28? No, 1/4 is equal to 7/28? Wait, 1/4 is equal to 7/28? Wait, 1/4 is equal to 7/28? Let me compute 7 divided by 28. That is 0.25, which is 1/4. Yes, so 1/4 is 7/28. Similarly, 1/3 is equal to 7/21? Wait, 7 divided by 21 is 1/3. Yes. So, the inequality is 7/28 < x/7 < 7/21.

But maybe this isn't helpful. Alternatively, maybe I can write x/7 as a fraction with denominator 7. So, x/7 is just x over 7. So, if I want to find x such that 1/4 < x/7 < 1/3, then maybe I can find x such that when you divide by 7, it's between those two fractions.

Alternatively, maybe I can find a common denominator for 1/4 and 1/3. The least common denominator is 12. So, 1/4 is 3/12 and 1/3 is 4/12. Therefore, the inequality is 3/12 < x/7 < 4/12. But I don't know if that helps directly. Maybe I can cross-multiply?

Alternatively, maybe I can think of x as being between 7*(1/4) and 7*(1/3). Which is exactly what I did earlier. So, 7*(1/4) is 7/4 = 1.75 and 7*(1/3) is 7/3 ≈ 2.333...

So, again, x must be between 1.75 and 2.333..., so integer x is 2.

Alternatively, maybe I can think of fractions between 1/4 and 1/3. Let me think. For example, 1/4 is 0.25, 1/3 is approximately 0.333. So, what fractions between these two? Well, 2/7 is approximately 0.2857, which is between them. Is there another fraction? For example, 3/10 is 0.3, which is also between them. But the question is about x/7, so x must be such that when divided by 7, it's between 1/4 and 1/3. Therefore, x must be such that x/7 is equal to some fraction between those two. But since x is an integer, the only possible x is 2, because 2/7 is approximately 0.2857, which is between 0.25 and 0.333.

Alternatively, maybe I can check if there are other integers. For example, if x is 2, then 2/7 is between 1/4 and 1/3. If x is 1, then 1/7 is less than 1/4. If x is 3, then 3/7 is approximately 0.428, which is greater than 1/3. So, no other integers.

Wait, but let me check if there's another way. Maybe by converting the inequalities to decimals?

Original inequality: 1/4 is 0.25, 1/3 is approximately 0.3333. So, x/7 must be between 0.25 and 0.3333. Therefore, multiplying all parts by 7, we get x between 0.25*7 = 1.75 and 0.3333*7 ≈ 2.333. So again, x must be between 1.75 and 2.333. So, the only integer is 2.

Therefore, I think x is 2.

But let me check once more. Maybe I made a mistake in calculation? Let me verify:

If x = 2, then 2 divided by 7 is approximately 0.2857.

Now, 1/4 is 0.25, and 1/3 is approximately 0.3333. So 0.25 < 0.2857 < 0.3333. Yes, that's correct.

If x = 1, then 1/7 ≈ 0.1429, which is less than 0.25.

If x = 3, then 3/7 ≈ 0.4286, which is greater than 0.3333.

Therefore, there are no other integers.

Alternatively, maybe I can think of fractions with denominator 7. Let me think. The fractions with denominator 7 are 0/7, 1/7, 2/7, 3/7, etc. So, between 1/4 and 1/3, which is between 0.25 and 0.333..., so looking at the fractions with denominator 7:

1/7 ≈ 0.1429 (too small)

2/7 ≈ 0.2857 (between 0.25 and 0.333)

3/7 ≈ 0.4286 (too big)

So, only 2/7 is between 1/4 and 1/3. Therefore, x must be 2.

Therefore, the answer is 2.

I think that's thorough. I considered multiple approaches: solving the inequality algebraically, converting to decimals, checking fractions with denominator 7, and verifying by plugging in values. All methods lead to the conclusion that x = 2 is the only integer satisfying the inequality.

**Final Answer**
The integer $ x $ is \boxed{2}.

To solve the inequality:

$$
\frac{1}{4} < \frac{x}{7} < \frac{1}{3}
$$

we aim to find an integer $ x $ that satisfies this condition.

---

### Step 1: Eliminate the Denominator

To simplify the inequality, we can multiply all parts by 7 (since 7 is positive, the inequality direction remains unchanged):

$$
7 \cdot \frac{1}{4} < x < 7 \cdot \frac{1}{3}
$$

$$
\frac{7}{4} < x < \frac{7}{3}
$$

Now compute the decimal values:

$$
\frac{7}{4} = 1.75 \quad \text{and} \quad \frac{7}{3} \approx 2.333\ldots
$$

So the inequality becomes:

$$
1.75 < x < 2.333\ldots
$$

---

### Step 2: Identify the Integer in the Interval

We are looking for an **integer** $ x $ that lies strictly between 1.75 and 2.333...

The only integer in this interval is:

$$
x = 2
$$

---

### Step 3: Verify the Solution

Let’s plug $ x = 2 $ back into the original inequality to confirm:

$$
\frac{1}{4} < \frac{2}{7} < \frac{1}{3}
$$

Convert to decimals for comparison:

- $ \frac{1}{4} = 0.25 $
- $ \frac{2}{7} \approx 0.2857 $
- $ \frac{1}{3} \approx 0.3333 $

So:

$$
0.25 < 0.2857 < 0.3333
$$

This confirms that $ x = 2 $ satisfies the inequality.

---

### Step 4: Check for Other Possible Integers

- $ x = 1 $: $ \frac{1}{7} \approx 0.1429 $ → too small
- $ x = 3 $: $ \frac{3}{7} \approx 0.4286 $ → too large

Thus, **no other integers** satisfy the inequality.

---

### Final Answer

$$
\boxed{2}
$$

Solution: Multiplying all expressions in the inequalities by $7$, we have $\frac74 < x < \frac73$. Since $\frac 74$ is between $1$ and $2$, and $\frac 73$ is between $2$ and $3$, the only integer $x$ between these two fractions is $\boxed{2}$.


Explanation:
Okay, let me try to figure out this problem. So, the question is: If t(x) = 3 - g(x) and g(x) = √x, then what is t(g(16))? Hmm, let me break this down step by step.

First, I need to understand what each part means. There's a function t(x) which is defined in terms of another function g(x). Specifically, t(x) is equal to 3 minus g(x). And then they tell me that g(x) is the square root of x. So, maybe I can substitute g(x) into t(x) to get an expression for t(x) in terms of x? Let me try that.

So, if t(x) = 3 - g(x), and g(x) is √x, then substituting that in, t(x) would be 3 - √x. Let me write that down:

t(x) = 3 - √x

Okay, so that seems straightforward. Now, the question is asking for t(g(16)). Hmm, so that means I need to first find g(16), and then plug that result into the function t. Let me make sure I understand function composition here.

So, t(g(16)) means that I first apply g to 16, and then apply t to the result of that. So, step one: compute g(16). Since g(x) is √x, then g(16) is √16. What's the square root of 16? Well, √16 is 4, right? Because 4 times 4 is 16. So, g(16) = 4.

Now, the next step is to compute t(g(16)), which is t(4) because g(16) is 4. So, now I need to find t(4). Since t(x) is 3 - √x, then t(4) would be 3 - √4. Let me calculate that.

√4 is 2, so 3 - 2 equals 1. Therefore, t(4) is 1. Therefore, t(g(16)) is 1.

Wait, let me check if I did that correctly. Maybe I should verify each step again.

First, confirm that g(x) is √x. Yes, the problem says g(x) = √x. Then t(x) is 3 - g(x), so substituting, t(x) = 3 - √x. Then, compute g(16). √16 is 4. Then, plug that into t(x): t(4) = 3 - √4 = 3 - 2 = 1. Seems correct.

Alternatively, maybe I can approach this without substituting first. Let me try another way. Since t(g(16)) is the same as t evaluated at g(16). So, if I first compute g(16) = √16 = 4, then t(4) = 3 - g(4). Wait, hold on, is that right? Wait, t(x) is 3 - g(x). So, t(4) is 3 - g(4). But earlier, I thought t(x) was 3 - √x. But if I use the original definition, t(x) = 3 - g(x), so if I compute t(4), it's 3 - g(4). But since g(4) is √4, which is 2, so 3 - 2 is 1. So, same result.

Alternatively, if I didn't substitute t(x) into 3 - √x, but instead just followed the functions step by step. So, t(g(16)) is t(√16) which is t(4), which is 3 - g(4) which is 3 - √4 which is 1. So, same answer.

So, either way, I get 1. Therefore, the answer should be 1.

But let me check if there's any chance I misread the problem. The problem says t(x) = 3 - g(x), and g(x) = √x. So, t is defined in terms of g. So, if I wanted to write t(x) in terms of x, it's 3 - √x. So, then t(g(16)) is 3 - √(g(16)). Wait, but that would be 3 - √(√16). Wait, hold on, is that another interpretation? Wait, no. Wait, hold on. Wait, is t(g(16)) equal to 3 - g(g(16))? Or is it t(g(16)) = 3 - g(g(16))? Wait, no. Wait, no. Let me think again.

Wait, t(x) is 3 - g(x). So, if I have t(g(16)), that is, the function t is being applied to the value g(16). So, t(g(16)) = 3 - g(g(16)). Wait, hold on, is that correct? Wait, no. Wait, no. Wait, if t(x) = 3 - g(x), then t(something) is 3 - g(something). So, if the something is g(16), then t(g(16)) is 3 - g(g(16)). Wait, so that would be different from my previous calculation. Wait, so which is it?

Hold on, this is a critical point. Let me clarify.

Original problem: t(x) = 3 - g(x). So, t is a function that takes an input x, and outputs 3 minus g(x). Therefore, if I want to compute t(g(16)), that is, input to t is g(16). Therefore, t(g(16)) = 3 - g(g(16)).

Wait, so that would be different from my previous calculation where I thought t(g(16)) was t(4) = 3 - √4. But according to this, it's 3 - g(g(16)) which is 3 - g(4). Which is the same as 3 - √4. Which is 1. So, same answer.

But maybe I confused myself here. Let me think again.

If t(x) = 3 - g(x), then t(anything) = 3 - g(anything). Therefore, t(g(16)) = 3 - g(g(16)). So, first compute g(16) which is 4, then compute g(4) which is 2, then 3 - 2 = 1. So, same result.

Alternatively, if I had written t(x) = 3 - √x, then t(g(16)) would be t(4) = 3 - √4 = 1. So, same answer.

Therefore, regardless of whether I substitute first or compute step by step, I get 1.

But maybe I need to check if there's a different interpretation. For example, if someone thought that t(g(16)) is t(g(16)) meaning t multiplied by g(16), but no, the notation t(g(16)) is standard function composition, meaning t of g of 16.

Alternatively, maybe someone could misread the problem as t(x) = 3 - g(x), and then g(x) = √x, so t(g(16)) is 3 - g(16). But that would be incorrect, because t(g(16)) is 3 - g(g(16)), not 3 - g(16).

Wait, but if someone thought that t(g(16)) is t evaluated at g(16), which is 3 - g(g(16)), but if someone mistakenly thought that t(g(16)) is 3 - g(16), that would be wrong. But in this case, since we have t(x) = 3 - g(x), so t(anything) is 3 - g(anything). Therefore, t(g(16)) is 3 - g(g(16)).

But in my first calculation, I thought of t(x) as 3 - √x, so t(g(16)) is 3 - √(g(16)) which is 3 - √(4) = 1. Which is the same as 3 - g(g(16)) because g(g(16)) is g(4) which is √4 = 2. So, 3 - 2 = 1.

Therefore, both approaches are consistent.

So, maybe the confusion is whether t(g(16)) is 3 - g(g(16)) or 3 - g(16). But according to the definition, it's the former.

But let me check with numbers. Suppose I have t(x) = 3 - g(x). Let me take a simple example. Let me say x = 16. Then, t(16) = 3 - g(16) = 3 - √16 = 3 - 4 = -1.

But if I want t(g(16)), that is, first compute g(16) = 4, then compute t(4) = 3 - g(4) = 3 - √4 = 3 - 2 = 1.

Alternatively, if someone thought that t(g(16)) is 3 - g(16), that would be 3 - 4 = -1, which is t(16), not t(g(16)). So, that's different.

Therefore, it's important to note that t(g(16)) is not t(16), but rather t evaluated at g(16).

Therefore, the correct answer is 1.

But just to make sure, let me try to write out all steps again:

Given:

t(x) = 3 - g(x)

g(x) = √x

Compute t(g(16)).

First, compute g(16):

g(16) = √16 = 4.

Now, compute t(g(16)) = t(4).

Now, compute t(4):

t(4) = 3 - g(4) = 3 - √4 = 3 - 2 = 1.

Therefore, the answer is 1.

Alternatively, if I substitute g(x) into t(x) first:

t(x) = 3 - g(x) = 3 - √x.

Therefore, t(g(16)) = t(√16) = t(4) = 3 - √4 = 1.

Same result.

Alternatively, if I think of t(g(x)) as a composition of functions, then t(g(x)) = 3 - g(g(x)) = 3 - √(√x) = 3 - x^{1/4}. But in this case, we are evaluating at x = 16, so t(g(16)) = 3 - √(√16) = 3 - √4 = 3 - 2 = 1.

So, again, same answer.

Therefore, regardless of the method, the answer is 1.

I think that's thorough. I considered different ways to approach the problem, checked for possible misunderstandings, and verified each step. So, I feel confident that the answer is 1.

**Final Answer**
\boxed{1}

To solve the problem, we are given two functions:

- $ t(x) = 3 - g(x) $
- $ g(x) = \sqrt{x} $

We are asked to compute $ t(g(16)) $.

---

### Step 1: Evaluate $ g(16) $

Since $ g(x) = \sqrt{x} $, we substitute $ x = 16 $:

$$
g(16) = \sqrt{16} = 4
$$

---

### Step 2: Use the result in $ t(x) $

Now that we know $ g(16) = 4 $, we substitute this into the function $ t(x) $:

$$
t(g(16)) = t(4)
$$

Recall that $ t(x) = 3 - g(x) $, so:

$$
t(4) = 3 - g(4)
$$

Now compute $ g(4) $:

$$
g(4) = \sqrt{4} = 2
$$

So:

$$
t(4) = 3 - 2 = 1
$$

---

### Final Answer

$$
\boxed{1}
$$

[Assistant]
Explanation:
Okay, let me try to figure out this problem. So, the question is: If t(x) = 3 - g(x) and g(x) = √x, then what is t(g(16))? Hmm, let me break this down step by step.

First, I need to understand what each part means. There's a function t(x) which is defined in terms of another function g(x). Specifically, t(x) is equal to 3 minus g(x). And then they tell me that g(x) is the square root of x. So, maybe I can substitute g(x) into t(x) to get an expression for t(x) in terms of x? Let me try that.

So, if t(x) = 3 - g(x), and g(x) is √x, then substituting that in, t(x) would be 3 - √x. Let me write that down:

t(x) = 3 - √x

Okay, so that seems straightforward. Now, the question is asking for t(g(16)). Hmm, so that means I need to first find g(16), and then plug that result into the function t. Let me make sure I understand function composition here.

So, t(g(16)) means that I first apply g to 16, and then apply t to the result of that. So, step one: compute g(16). Since g(x) is √x, then g(16) is √16. What's the square root of 16? Well, √16 is 4, right? Because 4 times 4 is 16. So, g(16) = 4.

Now, the next step is to compute t(g(16)), which is t(4) because g(16) is 4. So, now I need to find t(4). Since t(x) is 3 - √x, then t(4) would be 3 - √4. Let me calculate that.

√4 is 2, so 3 - 2 equals 1. Therefore, t(4) is 1. Therefore, t(g(16)) is 1.

Wait, let me check if I did that correctly. Maybe I should verify each step again.

First, confirm that g(x) is √x. Yes, the problem says g(x) = √x. Then t(x) is 3 - g(x), so substituting, t(x) = 3 - √x. Then, compute g(16). √16 is 4. Then, plug that into t(x): t(4) = 3 - √4 = 3 - 2 = 1. Seems correct.

Alternatively, maybe I can approach this without substituting first. Let me try another way. Since t(g(16)) is the same as t evaluated at g(16). So, if I first compute g(16) = √16 = 4, then t(4) = 3 - g(4). Wait, hold on, is that right? Wait, t(x) is 3 - g(x). So, t(4) is 3 - g(4). But earlier, I thought t(x) was 3 - √x. But if I use the original definition, t(x) = 3 - g(x), so if I compute t(4), it's 3 - g(4). But since g(4) is √4, which is 2, so 3 - 2 is 1. So, same result.

Alternatively, if I didn't substitute t(x) into 3 - √x, but instead just followed the functions step by step. So, t(g(16)) is t(√16) which is t(4), which is 3 - g(4) which is 3 - √4 which is 1. So, same answer.

So, either way, I get 1. Therefore, the answer should be 1.

But let me check if there's any chance I misread the problem. The problem says t(x) = 3 - g(x), and g(x) = √x. So, t is defined in terms of g. So, if I wanted to write t(x) in terms of x, it's 3 - √x. So, then t(g(16)) is 3 - √(g(16)). Wait, but that would be 3 - √(√16). Wait, hold on, is that another interpretation? Wait, no. Wait, hold on. Wait, is t(g(16)) equal to 3 - g(g(16))? Or is it t(g(16)) = 3 - g(g(16))? Wait, no. Wait, no. Let me think again.

Wait, t(x) is 3 - g(x). So, if I have t(g(16)), that is, the function t is being applied to the value g(16). So, t(g(16)) = 3 - g(g(16)). Wait, hold on, is that correct? Wait, no. Wait, no. Wait, if t(x) = 3 - g(x), then t(something) is 3 - g(something). So, if the something is g(16), then t(g(16)) is 3 - g(g(16)). Wait, so that would be different from my previous calculation. Wait, so which is it?

Hold on, this is a critical point. Let me clarify.

Original problem: t(x) = 3 - g(x). So, t is a function that takes an input x, and outputs 3 minus g(x). Therefore, if I want to compute t(g(16)), that is, input to t is g(16). Therefore, t(g(16)) = 3 - g(g(16)).

Wait, so that would be different from my previous calculation where I thought t(g(16)) was t(4) = 3 - √4. But according to this, it's 3 - g(g(16)) which is 3 - g(4). Which is the same as 3 - √4. Which is 1. So, same answer.

But maybe I confused myself here. Let me think again.

If t(x) = 3 - g(x), then t(anything) = 3 - g(anything). Therefore, t(g(16)) = 3 - g(g(16)). So, first compute g(16) which is 4, then compute g(4) which is 2, then 3 - 2 = 1. So, same result.

Alternatively, if I had written t(x) = 3 - √x, then t(g(16)) would be t(4) = 3 - √4 = 1. So, same answer.

Therefore, regardless of whether I substitute first or compute step by step, I get 1.

But maybe I need to check if there's a different interpretation. For example, if someone thought that t(g(16)) is t(g(16)) meaning t multiplied by g(16), but no, the notation t(g(16)) is standard function composition, meaning t of g of 16.

Alternatively, maybe someone could misread the problem as t(x) = 3 - g(x), and then g(x) = √x, so t(g(16)) is 3 - g(16). But that would be incorrect, because t(g(16)) is 3 - g(g(16)), not 3 - g(16).

Wait, but if someone thought that t(g(16)) is t evaluated at g(16), which is 3 - g(g(16)), but if someone mistakenly thought that t(g(16)) is 3 - g(16), that would be wrong. But in this case, since we have t(x) = 3 - g(x), so t(anything) is 3 - g(anything). Therefore, t(g(16)) is 3 - g(g(16)).

But in my first calculation, I thought of t(x) as 3 - √x, so t(g(16)) is 3 - √(g(16)) which is 3 - √(4) = 1. Which is the same as 3 - g(g(16)) because g(g(16)) is g(4) which is √4 = 2. So, 3 - 2 = 1.

Therefore, both approaches are consistent.

So, maybe the confusion is whether t(g(16)) is 3 - g(g(16)) or 3 - g(16). But according to the definition, it's the former.

But let me check with numbers. Suppose I have t(x) = 3 - g(x). Let me take a simple example. Let me say x = 16. Then, t(16) = 3 - g(16) = 3 - √16 = 3 - 4 = -1.

But if I want t(g(16)), that is, first compute g(16) = 4, then compute t(4) = 3 - g(4) = 3 - √4 = 3 - 2 = 1.

Alternatively, if someone thought that t(g(16)) is 3 - g(16), that would be 3 - 4 = -1, which is t(16), not t(g(16)). So, that's different.

Therefore, it's important to note that t(g(16)) is not t(16), but rather t evaluated at g(16).

Therefore, the correct answer is 1.

But just to make sure, let me try to write out all steps again:

Given:

t(x) = 3 - g(x)

g(x) = √x

Compute t(g(16)).

First, compute g(16):

g(16) = √16 = 4.

Now, compute t(g(16)) = t(4).

Now, compute t(4):

t(4) = 3 - g(4) = 3 - √4 = 3 - 2 = 1.

Therefore, the answer is 1.

Alternatively, if I substitute g(x) into t(x) first:

t(x) = 3 - g(x) = 3 - √x.

Therefore, t(g(16)) = t(√16) = t(4) = 3 - √4 = 1.

Same result.

Alternatively, if I think of t(g(x)) as a composition of functions, then t(g(x)) = 3 - g(g(x)) = 3 - √(√x) = 3 - x^{1/4}. But in this case, we are evaluating at x = 16, so t(g(16)) = 3 - √(√16) = 3 - √4 = 3 - 2 = 1.

So, again, same answer.

Therefore, regardless of the method, the answer is 1.

I think that's thorough. I considered different ways to approach the problem, checked for possible misunderstandings, and verified each step. So, I feel confident that the answer is 1.

**Final Answer**
\boxed{1}

To solve the problem, we are given two functions:

- $ t(x) = 3 - g(x) $
- $ g(x) = \sqrt{x} $

We are asked to compute $ t(g(16)) $.

---

### Step 1: Evaluate $ g(16) $

Since $ g(x) = \sqrt{x} $, we substitute $ x = 16 $:

$$
g(16) = \sqrt{16} = 4
$$

---

### Step 2: Use the result in $ t(x) $

Now that we know $ g(16) = 4 $, we substitute this into the function $ t(x) $:

$$
t(g(16)) = t(4)
$$

Recall that $ t(x) = 3 - g(x) $, so:

$$
t(4) = 3 - g(4)
$$

Now compute $ g(4) $:

$$
g(4) = \sqrt{4} = 2
$$

So:

$$
t(4) = 3 - 2 = 1
$$

---

### Final Answer

$$
\boxed{1}
$$

Solution: We have $g(16) = 4$, so $t(g(16)) = t(4) = 3- g(4) = 3-\sqrt{4} = 3-2 = \boxed{1}$.


Explanation:
Okay, so I need to simplify this algebraic expression: 4 times (3r³ + 5r - 6) minus 6 times (2r³ - r² + 4r). And then express it in the form Ar² + Br + C, where A, B, and C are integers. Alright, let me take it step by step.

First, I remember that to simplify expressions like this, I need to distribute the coefficients outside the parentheses to each term inside. So, let me start with the first part: 4(3r³ + 5r - 6).

Let me distribute the 4 to each term inside the first set of parentheses.

So, 4 multiplied by 3r³ is... let me calculate that. 4*3 is 12, so that term becomes 12r³.

Next term is 4 multiplied by 5r. 4*5 is 20, so that term is 20r.

Then the last term is 4 multiplied by -6. 4*(-6) is -24.

So, after distributing the 4, the first part becomes 12r³ + 20r - 24.

Now, moving on to the second part of the expression: -6(2r³ - r² + 4r). Wait, there's a minus sign in front of the 6, so I need to distribute that as well.

Let me make sure I handle the signs correctly. The expression is minus 6 times the entire parentheses. So, I can think of it as -6*(2r³) + (-6)*(-r²) + (-6)*(4r).

Calculating each term:

First term: -6*2r³. That's -12r³.

Second term: -6*(-r²). Multiplying two negatives gives a positive. So, that's +6r².

Third term: -6*(4r). That's -24r.

So, after distributing the -6, the second part becomes -12r³ + 6r² - 24r.

Now, I need to combine the two parts: the first part (12r³ + 20r - 24) and the second part (-12r³ + 6r² - 24r).

Let me write them out together:

12r³ + 20r - 24 -12r³ + 6r² - 24r.

Now, I need to combine like terms. Let me list all the terms:

- The r³ terms: 12r³ and -12r³.

- The r² terms: There's a +6r².

- The r terms: 20r and -24r.

- The constant term: -24.

Let me handle each type of term one by one.

Starting with the r³ terms: 12r³ - 12r³. That should be 0. Because 12 minus 12 is 0. So, those terms cancel out.

Next, the r² term: only 6r². So that stays as is.

Then the r terms: 20r - 24r. Let me compute that. 20 - 24 is -4, so that term is -4r.

Then the constant term is -24.

So putting it all together, after combining like terms, the expression simplifies to:

6r² - 4r - 24.

Wait, but the problem says to express it in the form Ar² + Br + C. Let me check if I did everything correctly. Let me go through the steps again to make sure I didn't make a mistake.

First part: 4*(3r³ + 5r -6) = 12r³ + 20r -24. That seems right. 4*3r³ is 12r³, 4*5r is 20r, 4*(-6) is -24. Correct.

Second part: -6*(2r³ - r² +4r). Let me check that again. Distribute the -6:

-6*2r³ = -12r³. Correct.

-6*(-r²) = +6r². Correct.

-6*4r = -24r. Correct. So that part is -12r³ +6r² -24r. Correct.

Then combining:

12r³ -12r³ cancels. Then +6r². Then 20r -24r is -4r. Then -24. So yes, 6r² -4r -24.

But wait, the problem says to express it in the form Ar² + Br + C. So, in this case, A is 6, B is -4, and C is -24.

But let me check if there's any mistake here. Let me verify by expanding again.

Alternatively, maybe I can check by plugging in a value for r and see if both the original expression and the simplified expression give the same result. Let me try r = 1.

Original expression: 4*(3(1)^3 +5(1) -6) -6*(2(1)^3 - (1)^2 +4(1)).

Calculating inside the first parentheses: 3(1) +5(1) -6 = 3 +5 -6 = 2. Multiply by 4: 4*2 = 8.

Inside the second parentheses: 2(1) -1 +4(1) = 2 -1 +4 = 5. Multiply by -6: -6*5 = -30.

So total expression: 8 -30 = -22.

Now, simplified expression: 6(1)^2 -4(1) -24 = 6 -4 -24 = -22. Same result. Good.

Let me try another value, say r = 2.

Original expression:

First part: 4*(3*(8) +5*(2) -6) = 4*(24 +10 -6) = 4*(28) = 112.

Second part: -6*(2*(8) - (4) +4*(2)) = -6*(16 -4 +8) = -6*(20) = -120.

Total expression: 112 -120 = -8.

Simplified expression: 6*(4) -4*(2) -24 = 24 -8 -24 = -8. Same result. Good.

Another check: r = 0.

Original expression:

First part: 4*(0 +0 -6) = 4*(-6) = -24.

Second part: -6*(0 -0 +0) = -6*(0) = 0.

Total expression: -24 + 0 = -24.

Simplified expression: 6*(0) -4*(0) -24 = -24. Correct.

So seems like my simplified expression is correct. Therefore, the answer is 6r² -4r -24.

But let me check if the problem says "express your answer in the form Ar² + Br + C". So, in this case, A is 6, B is -4, and C is -24. So, they are integers, which is required.

Wait, but maybe I need to write it as 6r² + (-4)r + (-24). But since they just want the form, it's okay to have negative coefficients.

Alternatively, maybe I can write it as 6r² - 4r -24. Which is the same thing.

So, I think that's the simplified form. Let me check once more if I combined all terms correctly.

Original expansion:

First part: 12r³ +20r -24.

Second part: -12r³ +6r² -24r.

Adding them together:

12r³ -12r³ = 0.

Then 6r².

Then 20r -24r = -4r.

Then -24.

Yes, that's correct. So, 6r² -4r -24.

Therefore, the answer is 6r² -4r -24.

But let me check if there's any possible mistake in the signs. For example, when distributing the -6, the second term was -r². So, -6 times -r² is +6r². Correct.

And the third term was -6*(4r) = -24r. Correct.

Yes, so combining with the first part's 20r, gives -4r.

Yes, that seems correct.

Therefore, I think the simplified expression is 6r² -4r -24.

So, in the form Ar² + Br + C, A is 6, B is -4, and C is -24.

Therefore, the answer is 6r² -4r -24.

**Final Answer**
The simplified expression is \boxed{6r^2 - 4r - 24}.

To simplify the expression:

$$
4(3r^3 + 5r - 6) - 6(2r^3 - r^2 + 4r)
$$

we follow a step-by-step process of **distributing** and **combining like terms**.

---

### Step 1: Distribute the coefficients

**First term:**
$$
4(3r^3 + 5r - 6) = 4 \cdot 3r^3 + 4 \cdot 5r + 4 \cdot (-6) = 12r^3 + 20r - 24
$$

**Second term:**
$$
-6(2r^3 - r^2 + 4r) = -6 \cdot 2r^3 + (-6) \cdot (-r^2) + (-6) \cdot 4r = -12r^3 + 6r^2 - 24r
$$

---

### Step 2: Combine the results

Now, combine the two expanded expressions:

$$
(12r^3 + 20r - 24) + (-12r^3 + 6r^2 - 24r)
$$

Group like terms:

- $ r^3 $: $ 12r^3 - 12r^3 = 0 $
- $ r^2 $: $ 6r^2 $
- $ r $: $ 20r - 24r = -4r $
- Constants: $ -24 $

---

### Step 3: Write the simplified expression

$$
6r^2 - 4r - 24
$$

This is now in the form $ Ar^2 + Br + C $, where:

- $ A = 6 $
- $ B = -4 $
- $ C = -24 $

---

### Final Answer

$$
\boxed{6r^2 - 4r - 24}
$$

[Assistant]
Explanation:
Okay, so I need to simplify this algebraic expression: 4 times (3r³ + 5r - 6) minus 6 times (2r³ - r² + 4r). And then express it in the form Ar² + Br + C, where A, B, and C are integers. Alright, let me take it step by step.

First, I remember that to simplify expressions like this, I need to distribute the coefficients outside the parentheses to each term inside. So, let me start with the first part: 4(3r³ + 5r - 6).

Let me distribute the 4 to each term inside the first set of parentheses.

So, 4 multiplied by 3r³ is... let me calculate that. 4*3 is 12, so that term becomes 12r³.

Next term is 4 multiplied by 5r. 4*5 is 20, so that term is 20r.

Then the last term is 4 multiplied by -6. 4*(-6) is -24.

So, after distributing the 4, the first part becomes 12r³ + 20r - 24.

Now, moving on to the second part of the expression: -6(2r³ - r² + 4r). Wait, there's a minus sign in front of the 6, so I need to distribute that as well.

Let me make sure I handle the signs correctly. The expression is minus 6 times the entire parentheses. So, I can think of it as -6*(2r³) + (-6)*(-r²) + (-6)*(4r).

Calculating each term:

First term: -6*2r³. That's -12r³.

Second term: -6*(-r²). Multiplying two negatives gives a positive. So, that's +6r².

Third term: -6*(4r). That's -24r.

So, after distributing the -6, the second part becomes -12r³ + 6r² - 24r.

Now, I need to combine the two parts: the first part (12r³ + 20r - 24) and the second part (-12r³ + 6r² - 24r).

Let me write them out together:

12r³ + 20r - 24 -12r³ + 6r² - 24r.

Now, I need to combine like terms. Let me list all the terms:

- The r³ terms: 12r³ and -12r³.

- The r² terms: There's a +6r².

- The r terms: 20r and -24r.

- The constant term: -24.

Let me handle each type of term one by one.

Starting with the r³ terms: 12r³ - 12r³. That should be 0. Because 12 minus 12 is 0. So, those terms cancel out.

Next, the r² term: only 6r². So that stays as is.

Then the r terms: 20r - 24r. Let me compute that. 20 - 24 is -4, so that term is -4r.

Then the constant term is -24.

So putting it all together, after combining like terms, the expression simplifies to:

6r² - 4r - 24.

Wait, but the problem says to express it in the form Ar² + Br + C. Let me check if I did everything correctly. Let me go through the steps again to make sure I didn't make a mistake.

First part: 4*(3r³ + 5r -6) = 12r³ + 20r -24. That seems right. 4*3r³ is 12r³, 4*5r is 20r, 4*(-6) is -24. Correct.

Second part: -6*(2r³ - r² +4r). Let me check that again. Distribute the -6:

-6*2r³ = -12r³. Correct.

-6*(-r²) = +6r². Correct.

-6*4r = -24r. Correct. So that part is -12r³ +6r² -24r. Correct.

Then combining:

12r³ -12r³ cancels. Then +6r². Then 20r -24r is -4r. Then -24. So yes, 6r² -4r -24.

But wait, the problem says to express it in the form Ar² + Br + C. So, in this case, A is 6, B is -4, and C is -24.

But let me check if there's any mistake here. Let me verify by expanding again.

Alternatively, maybe I can check by plugging in a value for r and see if both the original expression and the simplified expression give the same result. Let me try r = 1.

Original expression: 4*(3(1)^3 +5(1) -6) -6*(2(1)^3 - (1)^2 +4(1)).

Calculating inside the first parentheses: 3(1) +5(1) -6 = 3 +5 -6 = 2. Multiply by 4: 4*2 = 8.

Inside the second parentheses: 2(1) -1 +4(1) = 2 -1 +4 = 5. Multiply by -6: -6*5 = -30.

So total expression: 8 -30 = -22.

Now, simplified expression: 6(1)^2 -4(1) -24 = 6 -4 -24 = -22. Same result. Good.

Let me try another value, say r = 2.

Original expression:

First part: 4*(3*(8) +5*(2) -6) = 4*(24 +10 -6) = 4*(28) = 112.

Second part: -6*(2*(8) - (4) +4*(2)) = -6*(16 -4 +8) = -6*(20) = -120.

Total expression: 112 -120 = -8.

Simplified expression: 6*(4) -4*(2) -24 = 24 -8 -24 = -8. Same result. Good.

Another check: r = 0.

Original expression:

First part: 4*(0 +0 -6) = 4*(-6) = -24.

Second part: -6*(0 -0 +0) = -6*(0) = 0.

Total expression: -24 + 0 = -24.

Simplified expression: 6*(0) -4*(0) -24 = -24. Correct.

So seems like my simplified expression is correct. Therefore, the answer is 6r² -4r -24.

But let me check if the problem says "express your answer in the form Ar² + Br + C". So, in this case, A is 6, B is -4, and C is -24. So, they are integers, which is required.

Wait, but maybe I need to write it as 6r² + (-4)r + (-24). But since they just want the form, it's okay to have negative coefficients.

Alternatively, maybe I can write it as 6r² - 4r -24. Which is the same thing.

So, I think that's the simplified form. Let me check once more if I combined all terms correctly.

Original expansion:

First part: 12r³ +20r -24.

Second part: -12r³ +6r² -24r.

Adding them together:

12r³ -12r³ = 0.

Then 6r².

Then 20r -24r = -4r.

Then -24.

Yes, that's correct. So, 6r² -4r -24.

Therefore, the answer is 6r² -4r -24.

But let me check if there's any possible mistake in the signs. For example, when distributing the -6, the second term was -r². So, -6 times -r² is +6r². Correct.

And the third term was -6*(4r) = -24r. Correct.

Yes, so combining with the first part's 20r, gives -4r.

Yes, that seems correct.

Therefore, I think the simplified expression is 6r² -4r -24.

So, in the form Ar² + Br + C, A is 6, B is -4, and C is -24.

Therefore, the answer is 6r² -4r -24.

**Final Answer**
The simplified expression is \boxed{6r^2 - 4r - 24}.

To simplify the expression:

$$
4(3r^3 + 5r - 6) - 6(2r^3 - r^2 + 4r)
$$

we follow a step-by-step process of **distributing** and **combining like terms**.

---

### Step 1: Distribute the coefficients

**First term:**
$$
4(3r^3 + 5r - 6) = 4 \cdot 3r^3 + 4 \cdot 5r + 4 \cdot (-6) = 12r^3 + 20r - 24
$$

**Second term:**
$$
-6(2r^3 - r^2 + 4r) = -6 \cdot 2r^3 + (-6) \cdot (-r^2) + (-6) \cdot 4r = -12r^3 + 6r^2 - 24r
$$

---

### Step 2: Combine the results

Now, combine the two expanded expressions:

$$
(12r^3 + 20r - 24) + (-12r^3 + 6r^2 - 24r)
$$

Group like terms:

- $ r^3 $: $ 12r^3 - 12r^3 = 0 $
- $ r^2 $: $ 6r^2 $
- $ r $: $ 20r - 24r = -4r $
- Constants: $ -24 $

---

### Step 3: Write the simplified expression

$$
6r^2 - 4r - 24
$$

This is now in the form $ Ar^2 + Br + C $, where:

- $ A = 6 $
- $ B = -4 $
- $ C = -24 $

---

### Final Answer

$$
\boxed{6r^2 - 4r - 24}
$$

Solution: Using the distributive property and combining like terms, we have  $4(3r^3+5r-6)-6(2r^3-r^2+4r) = 12r^3+20r-24-12r^3+6r^2-24r.$ Simplifying, we get $\boxed{6r^2-4r-24}.$