Explanation:
Okay, so I need to find the minimum value of the function f(x) = |x - p| + |x - 15| + |x - p - 15| where 0 < p < 15, and x is between p and 15. Hmm, let me think about how to approach this.

First, I remember that functions involving absolute values can often be minimized by considering the points where the expressions inside the absolute values change their sign, which are called critical points. These critical points are usually the points where the argument of the absolute value is zero. So, for each absolute value term, I can find where it changes its behavior.

Let me write down the three absolute value terms:

1. |x - p|
2. |x - 15|
3. |x - p - 15|

So, the critical points for each term are:

1. x = p
2. x = 15
3. x = p + 15

But wait, since 0 < p < 15, then p + 15 is definitely greater than 15. So, the critical points are at x = p, x = 15, and x = p + 15. However, our domain for x is between p and 15. So, the critical points within this interval are x = p and x = 15. The third critical point, x = p + 15, is outside of our interval of interest.

Therefore, within the interval [p, 15], the function f(x) can be broken down into different cases depending on whether x is less than or greater than p, but since our interval starts at p, maybe we need to check if there are any other critical points inside [p, 15]. Wait, but the only critical points in the entire real line are p, 15, and p + 15. Since p + 15 is outside the interval [p, 15], the only critical points within [p, 15] are p and 15.

But maybe I need to check if there's a point inside [p, 15] where the slope of f(x) changes? Wait, for absolute value functions, the slope changes at the critical points. So, if I can figure out the slope of f(x) in different intervals, I can find where the minimum occurs.

Alternatively, since the function is a sum of absolute values, the minimum occurs at a point where the number of terms with positive slope equals the number with negative slope. But I need to recall the exact method.

Alternatively, maybe I can think of f(x) as the sum of distances from x to three points: p, 15, and p + 15. Wait, but since p is between 0 and 15, p + 15 is between 15 and 30. So, the three points are p (between 0 and 15), 15, and p + 15 (between 15 and 30).

But we are looking for the minimum of the sum of distances from x to these three points, where x is restricted to [p, 15].

In general, the sum of distances from a point to several points is minimized at the median of those points. However, since we are restricted to an interval, the minimum might be at the median or at the boundary of the interval.

But let me think. The median of the three points p, 15, and p + 15. Let me order them. Since p is between 0 and 15, p < 15 < p + 15. So the three points in order are p, 15, p + 15. The median is 15. So, if there were no restrictions on x, the minimum would be at x = 15. However, our interval is [p, 15], so 15 is the upper bound. Therefore, if the median is within the interval, then the minimum is at the median. But since the median is 15, which is the endpoint of the interval, then the minimum is achieved at x = 15.

But wait, is that correct? Let me check with an example. Let me take p = 5, so 0 < 5 < 15. Then the three points are 5, 15, 20. The median is 15. So, if I compute f(x) at x = 15, it would be |15 - 5| + |15 - 15| + |15 - 20| = 10 + 0 + 5 = 15. If I take x = 5, which is the lower bound, then f(5) = |5 - 5| + |5 - 15| + |5 - 20| = 0 + 10 + 15 = 25. If I take x somewhere in between, say x = 10. Then f(10) = |10 - 5| + |10 - 15| + |10 - 20| = 5 + 5 + 10 = 20. So, indeed, the minimum seems to be at x = 15.

But wait, let me check another point. Suppose x is somewhere between p and 15. Let me take p = 5 again, and x = 15. But maybe if I take p closer to 15? Let me try p = 10. Then the three points are 10, 15, 25. The median is 15. So, f(15) = |15 - 10| + |15 - 15| + |15 - 25| = 5 + 0 + 10 = 15. If I take x = 10, f(10) = 0 + 5 + 15 = 20. If I take x = 12, f(12) = |12 - 10| + |12 - 15| + |12 - 25| = 2 + 3 + 13 = 18. So again, the minimum is at x = 15.

But wait, maybe there's a case where the minimum is not at x = 15? Let me think. Suppose p is very small, like p approaches 0. Let me take p approaching 0. Then the three points are approaching 0, 15, and 15. Wait, p + 15 approaches 15. So, the three points are p, 15, p + 15. If p approaches 0, then p + 15 approaches 15, so the three points are approaching 0, 15, 15. So, the median is 15. So, again, the minimum would be at x = 15.

But maybe if the median is not in the interval? Wait, but in our case, the median is always 15, which is the upper bound of the interval. So, if the median is at the upper bound, then the minimum is achieved at the upper bound.

But let me think again. Suppose we have three points, and we are restricted to an interval that includes the median. Then the minimum is at the median. If the median is outside the interval, then the minimum is at the closest endpoint. But in our case, the median is 15, which is the upper bound of the interval. Therefore, the minimum is achieved at x = 15.

But wait, let me check with another example. Let me take p = 7. Then the three points are 7, 15, 22. The median is 15. So, f(15) = |15 - 7| + |15 - 15| + |15 - 22| = 8 + 0 + 7 = 15. If I take x = 7, f(7) = 0 + 8 + 15 = 23. If I take x = 10, f(10) = 3 + 5 + 12 = 20. So again, the minimum is at x = 15.

But wait, maybe there's a case where the minimum is not at x = 15? Let me think. Suppose that the median is not in the interval. Wait, but in our case, the median is always 15, which is the upper bound. So, if the interval was [a, b], and the median is at b, then the minimum is at b.

But maybe I need to verify this more formally. Let me try to analyze the function f(x) on the interval [p, 15].

First, let's note that f(x) is the sum of three absolute values. Let me write f(x) as:

f(x) = |x - p| + |x - 15| + |x - (p + 15)|

Since 0 < p < 15, then p + 15 is between 15 and 30. So, for x in [p, 15], the term |x - (p + 15)| is equal to (p + 15) - x, because x ≤ 15 < p + 15. Similarly, |x - p| is x - p, since x ≥ p. And |x - 15| is 15 - x if x ≤ 15, which it is.

Therefore, for x in [p, 15], f(x) can be rewritten as:

f(x) = (x - p) + (15 - x) + (p + 15 - x)

Let me simplify this:

First term: x - p

Second term: 15 - x

Third term: p + 15 - x

Adding them together:

(x - p) + (15 - x) + (p + 15 - x)

Let me compute term by term:

x - p + 15 - x + p + 15 - x

Combine like terms:

x - x - x = -x

-p + p = 0

15 + 15 = 30

So overall:

- x + 30

Therefore, f(x) simplifies to 30 - x for x in [p, 15].

Wait, that seems surprising. So, if I simplify f(x) on [p, 15], it's just 30 - x? Then, since x is between p and 15, the function f(x) is linear decreasing from x = p to x = 15. Therefore, the minimum value would be at x = 15, which is 30 - 15 = 15.

But wait, in my earlier examples, when p = 5, f(15) was 15, which matches. When p = 10, f(15) was 15. So, seems like the minimum is 15.

But let me check again with p = 5 and x = 15.

Original function: |15 - 5| + |15 - 15| + |15 - 20| = 10 + 0 + 5 = 15.

But according to the simplified expression, 30 - x = 30 - 15 = 15. Correct.

Similarly, if I take x = p, which is 5, then f(x) = 30 - 5 = 25. Which matches the earlier calculation.

So, according to this, f(x) is linear decreasing on [p, 15], so the minimum is at x = 15, and the minimum value is 15.

But wait, that seems too straightforward. Let me check if my simplification is correct.

Original f(x) = |x - p| + |x - 15| + |x - p - 15|

For x in [p, 15], we have:

|x - p| = x - p (since x ≥ p)

|x - 15| = 15 - x (since x ≤ 15)

|x - p - 15| = |x - (p + 15)|. Since x ≤ 15 and p + 15 > 15, this is equal to (p + 15) - x.

Therefore, adding them up:

(x - p) + (15 - x) + (p + 15 - x)

= x - p + 15 - x + p + 15 - x

= (x - x - x) + (-p + p) + (15 + 15)

= (-x) + 0 + 30

= 30 - x

Yes, that's correct. Therefore, f(x) = 30 - x on [p, 15]. Therefore, since this is a linear function with slope -1, it is decreasing on the interval. Therefore, the minimum occurs at the right endpoint, x = 15, and the minimum value is 30 - 15 = 15.

But wait, the problem says "Determine the minimum value taken by f(x) for x in the interval p ≤ x ≤ 15." So, according to this, the answer is 15? But let me check again with another example. Let me take p = 3, so the three points are 3, 15, 18. Then f(x) on [3, 15] is 30 - x. So, at x = 15, f(x) = 15. Let me compute directly:

f(15) = |15 - 3| + |15 - 15| + |15 - 18| = 12 + 0 + 3 = 15. Correct.

But let me check another point. Let me take x = 10. Then f(10) = |10 - 3| + |10 - 15| + |10 - 18| = 7 + 5 + 8 = 20. Which is 30 - 10 = 20. Correct.

So, seems like the function is indeed 30 - x on [p, 15], so the minimum is 15.

But wait, the problem says "where 0 < p < 15". So, regardless of p, the minimum is 15? But let me think again. Suppose p approaches 15. Let me take p approaching 15 from below, say p = 15 - ε, where ε is a small positive number. Then, the three points are p = 15 - ε, 15, and p + 15 = 30 - ε. Then, f(x) on [p, 15] is 30 - x. So, at x = 15, f(x) = 30 - 15 = 15. But if p approaches 15, then the first term |x - p| becomes |x - 15 + ε|, but since x is between p and 15, which is between 15 - ε and 15. So, for x in [15 - ε, 15], |x - p| = |x - (15 - ε)| = x - (15 - ε) = x - 15 + ε. Then, |x - 15| = 15 - x. And |x - p - 15| = |x - (15 - ε) - 15| = |x - 30 + ε|. Since x ≤ 15, this is 30 - ε - x.

Adding them up: (x - 15 + ε) + (15 - x) + (30 - ε - x) = x -15 + ε + 15 - x + 30 - ε - x = (x - x - x) + (-15 + 15) + (ε - ε) + 30 = -x + 0 + 0 + 30 = 30 - x. So, same result. Therefore, even when p approaches 15, the function is still 30 - x. Therefore, the minimum is 15.

But wait, let me check with p = 14. Then, the three points are 14, 15, 29. Then, f(x) on [14, 15] is 30 - x. So, at x = 15, f(x) = 15. Let me compute directly:

f(15) = |15 - 14| + |15 - 15| + |15 - 29| = 1 + 0 + 14 = 15. Correct.

So, seems like regardless of p, the minimum value is 15.

But wait, the problem says "Determine the minimum value taken by f(x) for x in the interval p ≤ x ≤ 15." So, according to this, the answer is 15. But let me check again if there's a mistake in my reasoning.

Wait, but maybe I made a mistake in simplifying f(x). Let me check again.

Original function: f(x) = |x - p| + |x - 15| + |x - p - 15|

For x in [p, 15], we have:

|x - p| = x - p

|x - 15| = 15 - x

|x - p - 15| = |x - (p + 15)|. Since p + 15 > 15 and x ≤ 15, this is (p + 15) - x.

Therefore, adding them up:

(x - p) + (15 - x) + (p + 15 - x)

= x - p + 15 - x + p + 15 - x

= (x - x - x) + (-p + p) + (15 + 15)

= -x + 0 + 30

= 30 - x

Yes, that's correct. Therefore, f(x) = 30 - x on [p, 15]. Therefore, since this is linear decreasing, the minimum is at x = 15, which is 30 - 15 = 15.

But then why does the problem specify 0 < p < 15? Because if p were equal to 15, then the interval would be [15, 15], which is just a single point. But since p is strictly less than 15, the interval is [p, 15], which is valid.

But maybe there's a different approach? Let me think.

Alternatively, since f(x) is the sum of absolute values, maybe I can think of it as the sum of distances from x to three points. The sum of distances is minimized at the median. However, since we are restricted to the interval [p, 15], and the median of the three points p, 15, p + 15 is 15, which is the upper bound of the interval, then the minimum is achieved at 15.

Therefore, regardless of p, the minimum is 15.

But let me check with p = 0. Wait, p is greater than 0, but approaching 0. Let me take p approaching 0. Then, the three points are approaching 0, 15, 15. The median is 15. So, the minimum is at 15.

Alternatively, if p were 0, then the three points would be 0, 15, 15. Then, the sum of distances from x to these points would be |x - 0| + |x - 15| + |x - 15|. Which is |x| + 2|x - 15|. For x in [0, 15], this is x + 2(15 - x) = x + 30 - 2x = 30 - x. So, same as before. Therefore, the minimum is at x = 15, which is 15.

Therefore, even if p approaches 0, the minimum is still 15.

Therefore, regardless of p in (0, 15), the minimum value of f(x) on [p, 15] is 15.

But wait, the problem says "Determine the minimum value taken by f(x) for x in the interval p ≤ x ≤ 15." So, is the answer 15?

But let me check once more with another value. Let me take p = 2. Then, the three points are 2, 15, 17. Then, f(x) on [2, 15] is 30 - x. Therefore, at x = 15, f(x) = 15. Let me compute directly:

f(15) = |15 - 2| + |15 - 15| + |15 - 17| = 13 + 0 + 2 = 15. Correct.

Alternatively, if I take x = p = 2, f(2) = |2 - 2| + |2 - 15| + |2 - 17| = 0 + 13 + 15 = 28. Which is 30 - 2 = 28. Correct.

Therefore, all checks out.

But wait, the problem says "the minimum value taken by f(x) for x in the interval p ≤ x ≤ 15." So, according to this, the answer is 15. But maybe I need to check if there's a case where the minimum is lower?

Wait, suppose that the function f(x) is not linear? But according to the simplification, it is linear.

Wait, but let me think again. Suppose that there is a point in [p, 15] where the slope changes. Wait, but in our case, we have already considered the critical points. However, since we have simplified f(x) to 30 - x, which is linear, there are no changes in slope.

But why is that? Because when we have three absolute value terms, normally, the function would have different expressions in different intervals. However, in this case, because of the specific positions of the critical points relative to the interval [p, 15], all three absolute value terms can be simplified to linear expressions with coefficients that cancel out except for the -x term.

Therefore, the function is linear on the entire interval [p, 15], decreasing from 30 - p to 30 - 15 = 15. Therefore, the minimum is 15.

Therefore, the answer should be 15.

But let me check once more with p = 1. Then, the three points are 1, 15, 16. Then, f(x) on [1, 15] is 30 - x. Therefore, at x = 15, f(x) = 15. Let me compute directly:

f(15) = |15 - 1| + |15 - 15| + |15 - 16| = 14 + 0 + 1 = 15. Correct.

Therefore, I think my conclusion is correct. The minimum value is 15.

But wait, the problem says "Determine the minimum value taken by f(x) for x in the interval p ≤ x ≤ 15." So, maybe the answer is 15? But let me check if there's a different interpretation.

Wait, maybe I made a mistake in simplifying the absolute values? Let me check again.

Original function: f(x) = |x - p| + |x - 15| + |x - p - 15|

For x in [p, 15], we have:

- x - p ≥ 0, so |x - p| = x - p

- x - 15 ≤ 0, so |x - 15| = 15 - x

- x - p - 15 = x - (p + 15). Since x ≤ 15 and p + 15 > 15, this is negative. Therefore, |x - p - 15| = (p + 15) - x

Therefore, adding them up:

(x - p) + (15 - x) + (p + 15 - x)

= x - p + 15 - x + p + 15 - x

= (x - x - x) + (-p + p) + (15 + 15)

= -x + 0 + 30

= 30 - x

Yes, that's correct. Therefore, f(x) = 30 - x on [p, 15]. Therefore, the minimum is 15.

Therefore, the answer is 15.

But let me check if there's a different approach. For example, using calculus.

If I consider f(x) = 30 - x on [p, 15], then the derivative is -1, which is negative, so the function is decreasing. Therefore, the minimum is at x = 15.

Alternatively, if I didn't simplify, and tried to take derivative. But since f(x) is made of absolute values, the derivative would be piecewise. However, since we have already simplified it, calculus isn't necessary.

Alternatively, think about the function f(x) as the sum of distances. The sum of distances from x to three points. The median minimizes the sum of absolute deviations. However, since we are restricted to [p, 15], and the median is at 15, which is the upper bound, then the minimum is achieved at 15.

Therefore, regardless of p, the minimum is 15.

Therefore, I think the answer is 15.

But let me check once more with p = 7.5. Then, the three points are 7.5, 15, 22.5. Then, f(x) on [7.5, 15] is 30 - x. Therefore, at x = 15, f(x) = 15. Let me compute directly:

f(15) = |15 - 7.5| + |15 - 15| + |15 - 22.5| = 7.5 + 0 + 7.5 = 15. Correct.

Therefore, all examples confirm that the minimum value is 15.

Therefore, I think the answer is 15.

But wait, the problem says "the minimum value taken by f(x) for x in the interval p ≤ x ≤ 15." So, maybe the answer is 15? But why does the problem specify 0 < p < 15? Maybe to ensure that p + 15 is outside the interval?

Yes, because if p were equal to 15, then p + 15 = 30, but since p is less than 15, p + 15 is greater than 15, so outside the interval. Therefore, the simplification holds.

Therefore, I think the answer is 15.

But let me check once again with p = 1. Let me take x = 15. Then f(x) = |15 - 1| + |15 - 15| + |15 - 1 - 15| = |14| + 0 + | -1 | = 14 + 0 + 1 = 15. Correct.

Therefore, I think I can be confident that the minimum value is 15.

**Final Answer**
\boxed{15}

To determine the minimum value of the function
$$
f(x) = |x - p| + |x - 15| + |x - p - 15|
$$
for $ x \in [p, 15] $, where $ 0 < p < 15 $, we begin by analyzing the structure of the function.

---

### Step 1: Understand the Function

The function is a sum of three absolute value terms:

- $ |x - p| $
- $ |x - 15| $
- $ |x - (p + 15)| $

We are to minimize this function over the interval $ [p, 15] $, where $ 0 < p < 15 $.

---

### Step 2: Analyze the Behavior of Each Term in the Interval

Let’s examine the sign of each term within the interval $ [p, 15] $:

- $ x \geq p $, so $ |x - p| = x - p $
- $ x \leq 15 $, so $ |x - 15| = 15 - x $
- $ x \leq 15 < p + 15 $, so $ |x - (p + 15)| = (p + 15) - x $

Thus, within the interval $ [p, 15] $, the function becomes:

$$
f(x) = (x - p) + (15 - x) + (p + 15 - x)
$$

---

### Step 3: Simplify the Expression

Simplify the expression:

$$
f(x) = x - p + 15 - x + p + 15 - x
$$

Group like terms:

- $ x - x - x = -x $
- $ -p + p = 0 $
- $ 15 + 15 = 30 $

So, the function simplifies to:

$$
f(x) = 30 - x
$$

---

### Step 4: Analyze the Simplified Function

Now, we have:

$$
f(x) = 30 - x \quad \text{for } x \in [p, 15]
$$

This is a **linear function** with a **negative slope** (i.e., decreasing). Therefore, the **minimum value** occurs at the **right endpoint** of the interval, which is $ x = 15 $.

---

### Step 5: Compute the Minimum Value

Substitute $ x = 15 $ into the simplified function:

$$
f(15) = 30 - 15 = 15
$$

---

### Final Answer

$$
\boxed{15}
$$

[Assistant]
Explanation:
Okay, so I need to find the minimum value of the function f(x) = |x - p| + |x - 15| + |x - p - 15| where 0 < p < 15, and x is between p and 15. Hmm, let me think about how to approach this.

First, I remember that functions involving absolute values can often be minimized by considering the points where the expressions inside the absolute values change their sign, which are called critical points. These critical points are usually the points where the argument of the absolute value is zero. So, for each absolute value term, I can find where it changes its behavior.

Let me write down the three absolute value terms:

1. |x - p|
2. |x - 15|
3. |x - p - 15|

So, the critical points for each term are:

1. x = p
2. x = 15
3. x = p + 15

But wait, since 0 < p < 15, then p + 15 is definitely greater than 15. So, the critical points are at x = p, x = 15, and x = p + 15. However, our domain for x is between p and 15. So, the critical points within this interval are x = p and x = 15. The third critical point, x = p + 15, is outside of our interval of interest.

Therefore, within the interval [p, 15], the function f(x) can be broken down into different cases depending on whether x is less than or greater than p, but since our interval starts at p, maybe we need to check if there are any other critical points inside [p, 15]. Wait, but the only critical points in the entire real line are p, 15, and p + 15. Since p + 15 is outside the interval [p, 15], the only critical points within [p, 15] are p and 15.

But maybe I need to check if there's a point inside [p, 15] where the slope of f(x) changes? Wait, for absolute value functions, the slope changes at the critical points. So, if I can figure out the slope of f(x) in different intervals, I can find where the minimum occurs.

Alternatively, since the function is a sum of absolute values, the minimum occurs at a point where the number of terms with positive slope equals the number with negative slope. But I need to recall the exact method.

Alternatively, maybe I can think of f(x) as the sum of distances from x to three points: p, 15, and p + 15. Wait, but since p is between 0 and 15, p + 15 is between 15 and 30. So, the three points are p (between 0 and 15), 15, and p + 15 (between 15 and 30).

But we are looking for the minimum of the sum of distances from x to these three points, where x is restricted to [p, 15].

In general, the sum of distances from a point to several points is minimized at the median of those points. However, since we are restricted to an interval, the minimum might be at the median or at the boundary of the interval.

But let me think. The median of the three points p, 15, and p + 15. Let me order them. Since p is between 0 and 15, p < 15 < p + 15. So the three points in order are p, 15, p + 15. The median is 15. So, if there were no restrictions on x, the minimum would be at x = 15. However, our interval is [p, 15], so 15 is the upper bound. Therefore, if the median is within the interval, then the minimum is at the median. But since the median is 15, which is the endpoint of the interval, then the minimum is achieved at x = 15.

But wait, is that correct? Let me check with an example. Let me take p = 5, so 0 < 5 < 15. Then the three points are 5, 15, 20. The median is 15. So, if I compute f(x) at x = 15, it would be |15 - 5| + |15 - 15| + |15 - 20| = 10 + 0 + 5 = 15. If I take x = 5, which is the lower bound, then f(5) = |5 - 5| + |5 - 15| + |5 - 20| = 0 + 10 + 15 = 25. If I take x somewhere in between, say x = 10. Then f(10) = |10 - 5| + |10 - 15| + |10 - 20| = 5 + 5 + 10 = 20. So, indeed, the minimum seems to be at x = 15.

But wait, let me check another point. Suppose x is somewhere between p and 15. Let me take p = 5 again, and x = 15. But maybe if I take p closer to 15? Let me try p = 10. Then the three points are 10, 15, 25. The median is 15. So, f(15) = |15 - 10| + |15 - 15| + |15 - 25| = 5 + 0 + 10 = 15. If I take x = 10, f(10) = 0 + 5 + 15 = 20. If I take x = 12, f(12) = |12 - 10| + |12 - 15| + |12 - 25| = 2 + 3 + 13 = 18. So again, the minimum is at x = 15.

But wait, maybe there's a case where the minimum is not at x = 15? Let me think. Suppose p is very small, like p approaches 0. Let me take p approaching 0. Then the three points are approaching 0, 15, and 15. Wait, p + 15 approaches 15. So, the three points are p, 15, p + 15. If p approaches 0, then p + 15 approaches 15, so the three points are approaching 0, 15, 15. So, the median is 15. So, again, the minimum would be at x = 15.

But maybe if the median is not in the interval? Wait, but in our case, the median is always 15, which is the upper bound of the interval. So, if the median is at the upper bound, then the minimum is achieved at the upper bound.

But let me think again. Suppose we have three points, and we are restricted to an interval that includes the median. Then the minimum is at the median. If the median is outside the interval, then the minimum is at the closest endpoint. But in our case, the median is 15, which is the upper bound of the interval. Therefore, the minimum is achieved at x = 15.

But wait, let me check with another example. Let me take p = 7. Then the three points are 7, 15, 22. The median is 15. So, f(15) = |15 - 7| + |15 - 15| + |15 - 22| = 8 + 0 + 7 = 15. If I take x = 7, f(7) = 0 + 8 + 15 = 23. If I take x = 10, f(10) = 3 + 5 + 12 = 20. So again, the minimum is at x = 15.

But wait, maybe there's a case where the minimum is not at x = 15? Let me think. Suppose that the median is not in the interval. Wait, but in our case, the median is always 15, which is the upper bound. So, if the interval was [a, b], and the median is at b, then the minimum is at b.

But maybe I need to verify this more formally. Let me try to analyze the function f(x) on the interval [p, 15].

First, let's note that f(x) is the sum of three absolute values. Let me write f(x) as:

f(x) = |x - p| + |x - 15| + |x - (p + 15)|

Since 0 < p < 15, then p + 15 is between 15 and 30. So, for x in [p, 15], the term |x - (p + 15)| is equal to (p + 15) - x, because x ≤ 15 < p + 15. Similarly, |x - p| is x - p, since x ≥ p. And |x - 15| is 15 - x if x ≤ 15, which it is.

Therefore, for x in [p, 15], f(x) can be rewritten as:

f(x) = (x - p) + (15 - x) + (p + 15 - x)

Let me simplify this:

First term: x - p

Second term: 15 - x

Third term: p + 15 - x

Adding them together:

(x - p) + (15 - x) + (p + 15 - x)

Let me compute term by term:

x - p + 15 - x + p + 15 - x

Combine like terms:

x - x - x = -x

-p + p = 0

15 + 15 = 30

So overall:

- x + 30

Therefore, f(x) simplifies to 30 - x for x in [p, 15].

Wait, that seems surprising. So, if I simplify f(x) on [p, 15], it's just 30 - x? Then, since x is between p and 15, the function f(x) is linear decreasing from x = p to x = 15. Therefore, the minimum value would be at x = 15, which is 30 - 15 = 15.

But wait, in my earlier examples, when p = 5, f(15) was 15, which matches. When p = 10, f(15) was 15. So, seems like the minimum is 15.

But let me check again with p = 5 and x = 15.

Original function: |15 - 5| + |15 - 15| + |15 - 20| = 10 + 0 + 5 = 15.

But according to the simplified expression, 30 - x = 30 - 15 = 15. Correct.

Similarly, if I take x = p, which is 5, then f(x) = 30 - 5 = 25. Which matches the earlier calculation.

So, according to this, f(x) is linear decreasing on [p, 15], so the minimum is at x = 15, and the minimum value is 15.

But wait, that seems too straightforward. Let me check if my simplification is correct.

Original f(x) = |x - p| + |x - 15| + |x - p - 15|

For x in [p, 15], we have:

|x - p| = x - p (since x ≥ p)

|x - 15| = 15 - x (since x ≤ 15)

|x - p - 15| = |x - (p + 15)|. Since x ≤ 15 and p + 15 > 15, this is equal to (p + 15) - x.

Therefore, adding them up:

(x - p) + (15 - x) + (p + 15 - x)

= x - p + 15 - x + p + 15 - x

= (x - x - x) + (-p + p) + (15 + 15)

= (-x) + 0 + 30

= 30 - x

Yes, that's correct. Therefore, f(x) = 30 - x on [p, 15]. Therefore, since this is a linear function with slope -1, it is decreasing on the interval. Therefore, the minimum occurs at the right endpoint, x = 15, and the minimum value is 30 - 15 = 15.

But wait, the problem says "Determine the minimum value taken by f(x) for x in the interval p ≤ x ≤ 15." So, according to this, the answer is 15? But let me check again with another example. Let me take p = 3, so the three points are 3, 15, 18. Then f(x) on [3, 15] is 30 - x. So, at x = 15, f(x) = 15. Let me compute directly:

f(15) = |15 - 3| + |15 - 15| + |15 - 18| = 12 + 0 + 3 = 15. Correct.

But let me check another point. Let me take x = 10. Then f(10) = |10 - 3| + |10 - 15| + |10 - 18| = 7 + 5 + 8 = 20. Which is 30 - 10 = 20. Correct.

So, seems like the function is indeed 30 - x on [p, 15], so the minimum is 15.

But wait, the problem says "where 0 < p < 15". So, regardless of p, the minimum is 15? But let me think again. Suppose p approaches 15. Let me take p approaching 15 from below, say p = 15 - ε, where ε is a small positive number. Then, the three points are p = 15 - ε, 15, and p + 15 = 30 - ε. Then, f(x) on [p, 15] is 30 - x. So, at x = 15, f(x) = 30 - 15 = 15. But if p approaches 15, then the first term |x - p| becomes |x - 15 + ε|, but since x is between p and 15, which is between 15 - ε and 15. So, for x in [15 - ε, 15], |x - p| = |x - (15 - ε)| = x - (15 - ε) = x - 15 + ε. Then, |x - 15| = 15 - x. And |x - p - 15| = |x - (15 - ε) - 15| = |x - 30 + ε|. Since x ≤ 15, this is 30 - ε - x.

Adding them up: (x - 15 + ε) + (15 - x) + (30 - ε - x) = x -15 + ε + 15 - x + 30 - ε - x = (x - x - x) + (-15 + 15) + (ε - ε) + 30 = -x + 0 + 0 + 30 = 30 - x. So, same result. Therefore, even when p approaches 15, the function is still 30 - x. Therefore, the minimum is 15.

But wait, let me check with p = 14. Then, the three points are 14, 15, 29. Then, f(x) on [14, 15] is 30 - x. So, at x = 15, f(x) = 15. Let me compute directly:

f(15) = |15 - 14| + |15 - 15| + |15 - 29| = 1 + 0 + 14 = 15. Correct.

So, seems like regardless of p, the minimum value is 15.

But wait, the problem says "Determine the minimum value taken by f(x) for x in the interval p ≤ x ≤ 15." So, according to this, the answer is 15. But let me check again if there's a mistake in my reasoning.

Wait, but maybe I made a mistake in simplifying f(x). Let me check again.

Original function: f(x) = |x - p| + |x - 15| + |x - p - 15|

For x in [p, 15], we have:

|x - p| = x - p

|x - 15| = 15 - x

|x - p - 15| = |x - (p + 15)|. Since p + 15 > 15 and x ≤ 15, this is (p + 15) - x.

Therefore, adding them up:

(x - p) + (15 - x) + (p + 15 - x)

= x - p + 15 - x + p + 15 - x

= (x - x - x) + (-p + p) + (15 + 15)

= -x + 0 + 30

= 30 - x

Yes, that's correct. Therefore, f(x) = 30 - x on [p, 15]. Therefore, since this is linear decreasing, the minimum is at x = 15, which is 30 - 15 = 15.

But then why does the problem specify 0 < p < 15? Because if p were equal to 15, then the interval would be [15, 15], which is just a single point. But since p is strictly less than 15, the interval is [p, 15], which is valid.

But maybe there's a different approach? Let me think.

Alternatively, since f(x) is the sum of absolute values, maybe I can think of it as the sum of distances from x to three points. The sum of distances is minimized at the median. However, since we are restricted to the interval [p, 15], and the median of the three points p, 15, p + 15 is 15, which is the upper bound of the interval, then the minimum is achieved at 15.

Therefore, regardless of p, the minimum is 15.

But let me check with p = 0. Wait, p is greater than 0, but approaching 0. Let me take p approaching 0. Then, the three points are approaching 0, 15, 15. The median is 15. So, the minimum is at 15.

Alternatively, if p were 0, then the three points would be 0, 15, 15. Then, the sum of distances from x to these points would be |x - 0| + |x - 15| + |x - 15|. Which is |x| + 2|x - 15|. For x in [0, 15], this is x + 2(15 - x) = x + 30 - 2x = 30 - x. So, same as before. Therefore, the minimum is at x = 15, which is 15.

Therefore, even if p approaches 0, the minimum is still 15.

Therefore, regardless of p in (0, 15), the minimum value of f(x) on [p, 15] is 15.

But wait, the problem says "Determine the minimum value taken by f(x) for x in the interval p ≤ x ≤ 15." So, is the answer 15?

But let me check once more with another value. Let me take p = 2. Then, the three points are 2, 15, 17. Then, f(x) on [2, 15] is 30 - x. Therefore, at x = 15, f(x) = 15. Let me compute directly:

f(15) = |15 - 2| + |15 - 15| + |15 - 17| = 13 + 0 + 2 = 15. Correct.

Alternatively, if I take x = p = 2, f(2) = |2 - 2| + |2 - 15| + |2 - 17| = 0 + 13 + 15 = 28. Which is 30 - 2 = 28. Correct.

Therefore, all checks out.

But wait, the problem says "the minimum value taken by f(x) for x in the interval p ≤ x ≤ 15." So, according to this, the answer is 15. But maybe I need to check if there's a case where the minimum is lower?

Wait, suppose that the function f(x) is not linear? But according to the simplification, it is linear.

Wait, but let me think again. Suppose that there is a point in [p, 15] where the slope changes. Wait, but in our case, we have already considered the critical points. However, since we have simplified f(x) to 30 - x, which is linear, there are no changes in slope.

But why is that? Because when we have three absolute value terms, normally, the function would have different expressions in different intervals. However, in this case, because of the specific positions of the critical points relative to the interval [p, 15], all three absolute value terms can be simplified to linear expressions with coefficients that cancel out except for the -x term.

Therefore, the function is linear on the entire interval [p, 15], decreasing from 30 - p to 30 - 15 = 15. Therefore, the minimum is 15.

Therefore, the answer should be 15.

But let me check once more with p = 1. Then, the three points are 1, 15, 16. Then, f(x) on [1, 15] is 30 - x. Therefore, at x = 15, f(x) = 15. Let me compute directly:

f(15) = |15 - 1| + |15 - 15| + |15 - 16| = 14 + 0 + 1 = 15. Correct.

Therefore, I think my conclusion is correct. The minimum value is 15.

But wait, the problem says "Determine the minimum value taken by f(x) for x in the interval p ≤ x ≤ 15." So, maybe the answer is 15? But let me check if there's a different interpretation.

Wait, maybe I made a mistake in simplifying the absolute values? Let me check again.

Original function: f(x) = |x - p| + |x - 15| + |x - p - 15|

For x in [p, 15], we have:

- x - p ≥ 0, so |x - p| = x - p

- x - 15 ≤ 0, so |x - 15| = 15 - x

- x - p - 15 = x - (p + 15). Since x ≤ 15 and p + 15 > 15, this is negative. Therefore, |x - p - 15| = (p + 15) - x

Therefore, adding them up:

(x - p) + (15 - x) + (p + 15 - x)

= x - p + 15 - x + p + 15 - x

= (x - x - x) + (-p + p) + (15 + 15)

= -x + 0 + 30

= 30 - x

Yes, that's correct. Therefore, f(x) = 30 - x on [p, 15]. Therefore, the minimum is 15.

Therefore, the answer is 15.

But let me check if there's a different approach. For example, using calculus.

If I consider f(x) = 30 - x on [p, 15], then the derivative is -1, which is negative, so the function is decreasing. Therefore, the minimum is at x = 15.

Alternatively, if I didn't simplify, and tried to take derivative. But since f(x) is made of absolute values, the derivative would be piecewise. However, since we have already simplified it, calculus isn't necessary.

Alternatively, think about the function f(x) as the sum of distances. The sum of distances from x to three points. The median minimizes the sum of absolute deviations. However, since we are restricted to [p, 15], and the median is at 15, which is the upper bound, then the minimum is achieved at 15.

Therefore, regardless of p, the minimum is 15.

Therefore, I think the answer is 15.

But let me check once more with p = 7.5. Then, the three points are 7.5, 15, 22.5. Then, f(x) on [7.5, 15] is 30 - x. Therefore, at x = 15, f(x) = 15. Let me compute directly:

f(15) = |15 - 7.5| + |15 - 15| + |15 - 22.5| = 7.5 + 0 + 7.5 = 15. Correct.

Therefore, all examples confirm that the minimum value is 15.

Therefore, I think the answer is 15.

But wait, the problem says "the minimum value taken by f(x) for x in the interval p ≤ x ≤ 15." So, maybe the answer is 15? But why does the problem specify 0 < p < 15? Maybe to ensure that p + 15 is outside the interval?

Yes, because if p were equal to 15, then p + 15 = 30, but since p is less than 15, p + 15 is greater than 15, so outside the interval. Therefore, the simplification holds.

Therefore, I think the answer is 15.

But let me check once again with p = 1. Let me take x = 15. Then f(x) = |15 - 1| + |15 - 15| + |15 - 1 - 15| = |14| + 0 + | -1 | = 14 + 0 + 1 = 15. Correct.

Therefore, I think I can be confident that the minimum value is 15.

**Final Answer**
\boxed{15}

To determine the minimum value of the function
$$
f(x) = |x - p| + |x - 15| + |x - p - 15|
$$
for $ x \in [p, 15] $, where $ 0 < p < 15 $, we begin by analyzing the structure of the function.

---

### Step 1: Understand the Function

The function is a sum of three absolute value terms:

- $ |x - p| $
- $ |x - 15| $
- $ |x - (p + 15)| $

We are to minimize this function over the interval $ [p, 15] $, where $ 0 < p < 15 $.

---

### Step 2: Analyze the Behavior of Each Term in the Interval

Let’s examine the sign of each term within the interval $ [p, 15] $:

- $ x \geq p $, so $ |x - p| = x - p $
- $ x \leq 15 $, so $ |x - 15| = 15 - x $
- $ x \leq 15 < p + 15 $, so $ |x - (p + 15)| = (p + 15) - x $

Thus, within the interval $ [p, 15] $, the function becomes:

$$
f(x) = (x - p) + (15 - x) + (p + 15 - x)
$$

---

### Step 3: Simplify the Expression

Simplify the expression:

$$
f(x) = x - p + 15 - x + p + 15 - x
$$

Group like terms:

- $ x - x - x = -x $
- $ -p + p = 0 $
- $ 15 + 15 = 30 $

So, the function simplifies to:

$$
f(x) = 30 - x
$$

---

### Step 4: Analyze the Simplified Function

Now, we have:

$$
f(x) = 30 - x \quad \text{for } x \in [p, 15]
$$

This is a **linear function** with a **negative slope** (i.e., decreasing). Therefore, the **minimum value** occurs at the **right endpoint** of the interval, which is $ x = 15 $.

---

### Step 5: Compute the Minimum Value

Substitute $ x = 15 $ into the simplified function:

$$
f(15) = 30 - 15 = 15
$$

---

### Final Answer

$$
\boxed{15}
$$

Solution: Since $0 < p \le x \le 15,$ the absolute values simplify to \[f(x) = (x-p) - (x-15) - (x-p-15) = -x+30.\]The value of this expression is minimized when $x=15,$ giving $-15+30=\boxed{15}.$


Explanation:
Okay, so I need to find the equation of the line that consists of all vectors v such that the projection of v onto the vector (2, 1) is the zero vector. Hmm, let me recall how projections work.

First, the projection of a vector v onto another vector a is given by the formula:

proj_a v = ( (v · a) / ||a||² ) * a

Right? So if the projection is the zero vector, that means that (v · a) / ||a||² * a = 0. Since a is not the zero vector, the scalar multiple must be zero. Therefore, (v · a) must be zero.

So, in this case, the projection of v onto (2, 1) is zero if and only if v is orthogonal to (2, 1). Therefore, the set of all such vectors v is the set of vectors orthogonal to (2, 1).

But wait, the problem says that these vectors lie on a line. But the set of vectors orthogonal to a non-zero vector in R² is a line through the origin, right? Because in two dimensions, the orthogonal complement of a one-dimensional subspace is another one-dimensional subspace, which is a line.

So, if I can find the equation of the line that consists of all vectors orthogonal to (2, 1), that should be the answer.

Let me write that down. Let v = (x, y). Then, the dot product of v and (2, 1) is 2x + y. For them to be orthogonal, this must equal zero. Therefore:

2x + y = 0

So, solving for y, we get y = -2x.

Wait, but the problem says "the equation of this line in the form y = mx + b". So, in this case, m is -2 and b is 0. Therefore, the equation is y = -2x.

But let me check if I did everything correctly. Let me verify with an example. Suppose v is (1, -2). Then, the projection onto (2, 1) would be [( (1)(2) + (-2)(1) ) / (2² + 1²)] * (2, 1) = [ (2 - 2)/5 ] * (2, 1) = 0 * (2, 1) = (0, 0). So that works.

Another example: take v = (0, 0). The projection is obviously zero. That's on the line y = -2x since 0 = -2*0.

What about a vector not on the line? Let's say (1, 0). The projection would be [ (2*1 + 1*0)/5 ]*(2, 1) = (2/5)(2, 1) which is not zero. So that's correct.

So, the line is y = -2x. Therefore, the equation is y = -2x + 0.

But wait, the problem says "the set of vectors v" lie on a line. But vectors are in R², so the line is through the origin? Because the equation y = -2x is a line through the origin.

But maybe the problem is expecting the answer in terms of coordinates? Wait, the question says "Enter the equation of this line in the form y = mx + b". So, since the line passes through the origin, b is zero.

Therefore, the answer is y = -2x.

But let me think again. The set of vectors v such that their projection onto (2, 1) is zero is the orthogonal complement of (2, 1). In R², that's a line. So, the equation is 2x + y = 0, which is equivalent to y = -2x.

Yes, that seems right.

Alternatively, if I think geometrically, the projection onto (2, 1) being zero means that the vector v is perpendicular to (2, 1). So, the line of all such vectors is the line perpendicular to (2, 1). The slope of (2, 1) is 1/2, so the slope of the perpendicular line is -2. Therefore, the line has slope -2 and passes through the origin, so y = -2x.

Yes, that's consistent.

Therefore, I think the answer is y = -2x.

**Final Answer**
The equation of the line is \boxed{y = -2x}.

To determine the equation of the line consisting of all vectors $\mathbf{v}$ such that

$$
\operatorname{proj}_{\begin{pmatrix} 2 \\ 1 \end{pmatrix}} \mathbf{v} = \begin{pmatrix} 0 \\ 0 \end{pmatrix},
$$

we begin by recalling the formula for the projection of a vector $\mathbf{v}$ onto another vector $\mathbf{a}$:

$$
\operatorname{proj}_{\mathbf{a}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{a}}{\|\mathbf{a}\|^2} \mathbf{a}.
$$

In this case, $\mathbf{a} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$. The projection is the zero vector if and only if the scalar factor is zero. That is:

$$
\frac{\mathbf{v} \cdot \mathbf{a}}{\|\mathbf{a}\|^2} = 0.
$$

Since $\|\mathbf{a}\|^2 = 2^2 + 1^2 = 5$, the condition becomes:

$$
\mathbf{v} \cdot \begin{pmatrix} 2 \\ 1 \end{pmatrix} = 0.
$$

Let $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}$. Then the dot product is:

$$
2x + y = 0.
$$

This is the equation of a line in the $xy$-plane. Solving for $y$, we get:

$$
y = -2x.
$$

---

### Interpretation

This line consists of all vectors $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}$ that are orthogonal to $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$. Geometrically, this is the line perpendicular to the vector $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$, which has slope $1/2$, so the perpendicular line has slope $-2$. Since the line passes through the origin (as the projection of the zero vector is zero), the equation is:

$$
y = -2x.
$$

---

### Final Answer

$$
\boxed{y = -2x}
$$

[Assistant]
Explanation:
Okay, so I need to find the equation of the line that consists of all vectors v such that the projection of v onto the vector (2, 1) is the zero vector. Hmm, let me recall how projections work.

First, the projection of a vector v onto another vector a is given by the formula:

proj_a v = ( (v · a) / ||a||² ) * a

Right? So if the projection is the zero vector, that means that (v · a) / ||a||² * a = 0. Since a is not the zero vector, the scalar multiple must be zero. Therefore, (v · a) must be zero.

So, in this case, the projection of v onto (2, 1) is zero if and only if v is orthogonal to (2, 1). Therefore, the set of all such vectors v is the set of vectors orthogonal to (2, 1).

But wait, the problem says that these vectors lie on a line. But the set of vectors orthogonal to a non-zero vector in R² is a line through the origin, right? Because in two dimensions, the orthogonal complement of a one-dimensional subspace is another one-dimensional subspace, which is a line.

So, if I can find the equation of the line that consists of all vectors orthogonal to (2, 1), that should be the answer.

Let me write that down. Let v = (x, y). Then, the dot product of v and (2, 1) is 2x + y. For them to be orthogonal, this must equal zero. Therefore:

2x + y = 0

So, solving for y, we get y = -2x.

Wait, but the problem says "the equation of this line in the form y = mx + b". So, in this case, m is -2 and b is 0. Therefore, the equation is y = -2x.

But let me check if I did everything correctly. Let me verify with an example. Suppose v is (1, -2). Then, the projection onto (2, 1) would be [( (1)(2) + (-2)(1) ) / (2² + 1²)] * (2, 1) = [ (2 - 2)/5 ] * (2, 1) = 0 * (2, 1) = (0, 0). So that works.

Another example: take v = (0, 0). The projection is obviously zero. That's on the line y = -2x since 0 = -2*0.

What about a vector not on the line? Let's say (1, 0). The projection would be [ (2*1 + 1*0)/5 ]*(2, 1) = (2/5)(2, 1) which is not zero. So that's correct.

So, the line is y = -2x. Therefore, the equation is y = -2x + 0.

But wait, the problem says "the set of vectors v" lie on a line. But vectors are in R², so the line is through the origin? Because the equation y = -2x is a line through the origin.

But maybe the problem is expecting the answer in terms of coordinates? Wait, the question says "Enter the equation of this line in the form y = mx + b". So, since the line passes through the origin, b is zero.

Therefore, the answer is y = -2x.

But let me think again. The set of vectors v such that their projection onto (2, 1) is zero is the orthogonal complement of (2, 1). In R², that's a line. So, the equation is 2x + y = 0, which is equivalent to y = -2x.

Yes, that seems right.

Alternatively, if I think geometrically, the projection onto (2, 1) being zero means that the vector v is perpendicular to (2, 1). So, the line of all such vectors is the line perpendicular to (2, 1). The slope of (2, 1) is 1/2, so the slope of the perpendicular line is -2. Therefore, the line has slope -2 and passes through the origin, so y = -2x.

Yes, that's consistent.

Therefore, I think the answer is y = -2x.

**Final Answer**
The equation of the line is \boxed{y = -2x}.

To determine the equation of the line consisting of all vectors $\mathbf{v}$ such that

$$
\operatorname{proj}_{\begin{pmatrix} 2 \\ 1 \end{pmatrix}} \mathbf{v} = \begin{pmatrix} 0 \\ 0 \end{pmatrix},
$$

we begin by recalling the formula for the projection of a vector $\mathbf{v}$ onto another vector $\mathbf{a}$:

$$
\operatorname{proj}_{\mathbf{a}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{a}}{\|\mathbf{a}\|^2} \mathbf{a}.
$$

In this case, $\mathbf{a} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$. The projection is the zero vector if and only if the scalar factor is zero. That is:

$$
\frac{\mathbf{v} \cdot \mathbf{a}}{\|\mathbf{a}\|^2} = 0.
$$

Since $\|\mathbf{a}\|^2 = 2^2 + 1^2 = 5$, the condition becomes:

$$
\mathbf{v} \cdot \begin{pmatrix} 2 \\ 1 \end{pmatrix} = 0.
$$

Let $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}$. Then the dot product is:

$$
2x + y = 0.
$$

This is the equation of a line in the $xy$-plane. Solving for $y$, we get:

$$
y = -2x.
$$

---

### Interpretation

This line consists of all vectors $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}$ that are orthogonal to $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$. Geometrically, this is the line perpendicular to the vector $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$, which has slope $1/2$, so the perpendicular line has slope $-2$. Since the line passes through the origin (as the projection of the zero vector is zero), the equation is:

$$
y = -2x.
$$

---

### Final Answer

$$
\boxed{y = -2x}
$$

Solution:
Let $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}.$

From the formula of a projection,
\begin{align*}
\operatorname{proj}_{\begin{pmatrix} 2 \\ 1 \end{pmatrix}} \mathbf{v} &= \frac{\mathbf{v} \cdot \begin{pmatrix} 2 \\ 1 \end{pmatrix}}{\left\| \begin{pmatrix} 2 \\ 1 \end{pmatrix} \right\|^2} \begin{pmatrix} 2 \\ 1 \end{pmatrix} \\
&= \frac{\begin{pmatrix} x \\ y \end{pmatrix} \cdot \begin{pmatrix} 2 \\ 1 \end{pmatrix}}{5} \begin{pmatrix} 2 \\ 1 \end{pmatrix} \\
&= \frac{2x + y}{5} \begin{pmatrix} 2 \\ 1 \end{pmatrix} \\
&= \begin{pmatrix} 0 \\ 0 \end{pmatrix}.
\end{align*}Then
\[\frac{2x + y}{5} = 0,\]so $2x + y = 0.$  Thus, the equation of the line is $\boxed{y = -2x}.$


Explanation:
Okay, so I need to find the modulus of this complex number w, which is given by (3z + 1) divided by (5z + 7), where z is 1 + i. Alright, let me start by recalling that the modulus of a complex number a + bi is sqrt(a² + b²). But first, I need to figure out what w actually is. Since z is given, maybe I should substitute z into the expression for w and simplify it first. Then, once I have w in the form a + bi, I can compute its modulus.

Let me write down z first: z = 1 + i. So, let me compute the numerator and the denominator separately. Let's start with the numerator: 3z + 1. Substituting z, that would be 3*(1 + i) + 1. Let me compute that:

3*(1 + i) is 3*1 + 3*i, which is 3 + 3i. Then adding 1 gives 3 + 1 + 3i, which is 4 + 3i. So the numerator is 4 + 3i.

Now the denominator: 5z + 7. Again, substituting z = 1 + i, so 5*(1 + i) + 7. Let me compute that:

5*(1 + i) is 5*1 + 5*i = 5 + 5i. Then adding 7 gives 5 + 7 + 5i, which is 12 + 5i. So the denominator is 12 + 5i.

Therefore, w is (4 + 3i)/(12 + 5i). Now, to simplify this complex fraction, I need to multiply the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of 12 + 5i is 12 - 5i. So, multiplying both numerator and denominator by 12 - 5i should eliminate the imaginary part in the denominator.

Let me write that out:

w = [(4 + 3i)(12 - 5i)] / [(12 + 5i)(12 - 5i)]

First, let me compute the denominator because that might be simpler. The denominator is (12 + 5i)(12 - 5i). This is a difference of squares, right? So it's 12² - (5i)². Let me compute that:

12² is 144. (5i)² is 25i², and since i² = -1, that becomes 25*(-1) = -25. Therefore, the denominator is 144 - (-25) = 144 + 25 = 169. Okay, so the denominator simplifies to 169. That's nice.

Now the numerator: (4 + 3i)(12 - 5i). Let me use the distributive property (FOIL method) to multiply these two complex numbers.

First, multiply 4 by 12: 4*12 = 48.

Then, 4*(-5i) = -20i.

Next, 3i*12 = 36i.

Finally, 3i*(-5i) = -15i².

So adding all these terms together: 48 - 20i + 36i -15i².

Combine like terms: the real parts are 48 and -15i², and the imaginary parts are -20i + 36i.

First, let me handle the imaginary parts: -20i + 36i = 16i.

Now, the real parts: 48 -15i². Remember that i² = -1, so -15i² is -15*(-1) = 15. Therefore, the real part is 48 + 15 = 63.

So altogether, the numerator simplifies to 63 + 16i.

Therefore, w = (63 + 16i)/169. Which can be written as (63/169) + (16/169)i.

Now, to find |w|, which is the modulus of w. Since w is a complex number a + bi, where a = 63/169 and b = 16/169, the modulus is sqrt(a² + b²).

So let me compute that:

|w| = sqrt[(63/169)² + (16/169)²] = sqrt[(63² + 16²)/169²] = sqrt(63² + 16²)/sqrt(169²) = sqrt(63² + 16²)/169.

So now I need to compute 63 squared plus 16 squared. Let me calculate each term:

First, 63 squared. Let me recall that 60² = 3600, 63² = (60 + 3)² = 60² + 2*60*3 + 3² = 3600 + 360 + 9 = 3600 + 360 is 3960, plus 9 is 3969.

Then 16 squared is 256. Therefore, adding them together: 3969 + 256. Let me compute that:

3969 + 200 = 4169, then +56 = 4225. Wait, 3969 + 256. Let me check again:

3969 + 256:

Breakdown 256 into 200 + 50 + 6.

3969 + 200 = 4169

4169 + 50 = 4219

4219 + 6 = 4225. Yes, that's correct. So 63² + 16² = 4225.

Therefore, |w| = sqrt(4225)/169. Now sqrt(4225)... Hmm, what is sqrt(4225)?

Well, I know that 65² = 4225 because 60² = 3600, 70² = 4900, so 65² is 4225. Let me check:

65*65: 60*60 + 60*5 + 5*60 + 5*5 = 3600 + 300 + 300 + 25 = 3600 + 600 + 25 = 4225. Yes, correct.

Therefore, sqrt(4225) is 65. Therefore, |w| = 65 / 169.

Now, simplifying 65/169. Let me see if 65 and 169 have a common factor. 65 factors into 5*13. 169 is 13². Therefore, 65 and 169 share a common factor of 13. So divide numerator and denominator by 13:

65 ÷13 = 5, and 169 ÷13 = 13. Therefore, 65/169 simplifies to 5/13.

Therefore, |w| = 5/13.

Wait, let me check that again. 65 divided by 13 is 5, yes. 169 divided by 13 is 13, yes. So 5/13. Therefore, the modulus of w is 5/13.

Let me just recap to make sure I didn't make any mistakes. Starting with z = 1 + i, computed numerator 3z + 1 as 4 + 3i, denominator 5z + 7 as 12 + 5i. Then multiplied numerator and denominator by 12 - 5i, got numerator as 63 + 16i and denominator 169. Then modulus squared was (63² + 16²)/169², which simplified to 4225/169², square root of that is 65/169 = 5/13. Seems correct.

Alternatively, maybe there's a smarter way to compute |w| without computing w explicitly? Let me think. Since |w| = |3z + 1| / |5z + 7|. Because modulus of a quotient is the quotient of the moduli. So maybe I could compute |3z + 1| and |5z + 7| separately and then divide them. Let me try that approach to verify.

First, compute |3z + 1|. Since z = 1 + i, 3z + 1 is 4 + 3i as before. The modulus of 4 + 3i is sqrt(4² + 3²) = sqrt(16 + 9) = sqrt(25) = 5. Okay, that's straightforward.

Then compute |5z + 7|. 5z + 7 is 12 + 5i as before. The modulus is sqrt(12² + 5²) = sqrt(144 + 25) = sqrt(169) = 13. Therefore, |w| = |3z + 1| / |5z + 7| = 5 / 13. Which matches the previous result. So that's a quicker way. So that confirms that the answer is 5/13.

Therefore, regardless of the method, I get 5/13. So I think that's the correct answer.

But just to make sure, let me check my steps again. First method: compute numerator and denominator, multiply by conjugate, get 63 + 16i over 169, then modulus squared is (63² + 16²)/169², which is 4225/28561, square root is 65/169 = 5/13. Second method: modulus of numerator is 5, modulus of denominator is 13, so 5/13. Both methods agree. So I think that's solid.

Alternatively, maybe I can check with another approach. Let me think. Suppose I didn't remember that |a/b| = |a|/|b|. But even so, the first method worked. So I think that's safe.

Alternatively, maybe I can compute w directly as a complex number and then compute modulus. Let me try that again. So w = (4 + 3i)/(12 + 5i). If I write this as (4 + 3i)/(12 + 5i), and then divide them, but maybe I can write it as a decimal or something? Probably not necessary, but just to check.

Alternatively, maybe I can write w in terms of real and imaginary parts. Wait, I already did that. So w = 63/169 + 16i/169. Then modulus squared is (63/169)^2 + (16/169)^2 = (63² + 16²)/169² = 4225/28561. Then sqrt(4225)/sqrt(28561) = 65/169 = 5/13. Yep, same result.

Alternatively, maybe I can check if 63 and 16 have any relation? Well, 63 is 7*9, 16 is 2^4. Not sure if that helps. But anyway, the key was that 63² + 16² = 4225, which is 65². So that's a key step. Let me check that again. 63² is 3969, 16² is 256. 3969 + 256. Let me add them again:

3969 + 256:

Start with 3969 + 200 = 4169

Then add 56: 4169 + 50 = 4219, then +6 = 4225. Correct. And 65² is 4225. So that's correct.

Therefore, all steps check out. Therefore, the modulus of w is 5/13.

I think that's thorough. I considered two different methods, checked my calculations multiple times, and everything seems consistent. So I feel confident that the answer is 5/13.

**Final Answer**
The modulus of $ w $ is \boxed{\dfrac{5}{13}}.

To find the modulus of the complex number $ w = \dfrac{3z + 1}{5z + 7} $, where $ z = 1 + i $, we proceed step by step.

---

### Step 1: Compute the Numerator and Denominator

We begin by substituting $ z = 1 + i $ into the expressions for the numerator and denominator.

- **Numerator**:
  $$
  3z + 1 = 3(1 + i) + 1 = 3 + 3i + 1 = 4 + 3i
  $$

- **Denominator**:
  $$
  5z + 7 = 5(1 + i) + 7 = 5 + 5i + 7 = 12 + 5i
  $$

So,
$$
w = \frac{4 + 3i}{12 + 5i}
$$

---

### Step 2: Simplify the Complex Fraction

To simplify the complex fraction, we multiply the numerator and denominator by the **complex conjugate** of the denominator, which is $ 12 - 5i $:

$$
w = \frac{(4 + 3i)(12 - 5i)}{(12 + 5i)(12 - 5i)}
$$

#### Compute the Denominator:

$$
(12 + 5i)(12 - 5i) = 12^2 - (5i)^2 = 144 - (-25) = 144 + 25 = 169
$$

#### Compute the Numerator:

Use the distributive property (FOIL):

$$
(4 + 3i)(12 - 5i) = 4 \cdot 12 + 4 \cdot (-5i) + 3i \cdot 12 + 3i \cdot (-5i)
$$
$$
= 48 - 20i + 36i - 15i^2 = 48 + 16i - 15(-1) = 48 + 16i + 15 = 63 + 16i
$$

So,
$$
w = \frac{63 + 16i}{169}
$$

---

### Step 3: Compute the Modulus of $ w $

The modulus of a complex number $ a + bi $ is given by:

$$
|w| = \sqrt{a^2 + b^2}
$$

Here, $ a = \frac{63}{169} $, $ b = \frac{16}{169} $, so:

$$
|w| = \sqrt{\left(\frac{63}{169}\right)^2 + \left(\frac{16}{169}\right)^2} = \frac{1}{169} \sqrt{63^2 + 16^2}
$$

Compute the squares:

- $ 63^2 = 3969 $
- $ 16^2 = 256 $

$$
|w| = \frac{1}{169} \sqrt{3969 + 256} = \frac{1}{169} \sqrt{4225}
$$

Now, $ \sqrt{4225} = 65 $, so:

$$
|w| = \frac{65}{169}
$$

Simplify the fraction:

$$
\frac{65}{169} = \frac{5}{13}
$$

---

### Final Answer

$$
\boxed{\dfrac{5}{13}}
$$

[Assistant]
Explanation:
Okay, so I need to find the modulus of this complex number w, which is given by (3z + 1) divided by (5z + 7), where z is 1 + i. Alright, let me start by recalling that the modulus of a complex number a + bi is sqrt(a² + b²). But first, I need to figure out what w actually is. Since z is given, maybe I should substitute z into the expression for w and simplify it first. Then, once I have w in the form a + bi, I can compute its modulus.

Let me write down z first: z = 1 + i. So, let me compute the numerator and the denominator separately. Let's start with the numerator: 3z + 1. Substituting z, that would be 3*(1 + i) + 1. Let me compute that:

3*(1 + i) is 3*1 + 3*i, which is 3 + 3i. Then adding 1 gives 3 + 1 + 3i, which is 4 + 3i. So the numerator is 4 + 3i.

Now the denominator: 5z + 7. Again, substituting z = 1 + i, so 5*(1 + i) + 7. Let me compute that:

5*(1 + i) is 5*1 + 5*i = 5 + 5i. Then adding 7 gives 5 + 7 + 5i, which is 12 + 5i. So the denominator is 12 + 5i.

Therefore, w is (4 + 3i)/(12 + 5i). Now, to simplify this complex fraction, I need to multiply the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of 12 + 5i is 12 - 5i. So, multiplying both numerator and denominator by 12 - 5i should eliminate the imaginary part in the denominator.

Let me write that out:

w = [(4 + 3i)(12 - 5i)] / [(12 + 5i)(12 - 5i)]

First, let me compute the denominator because that might be simpler. The denominator is (12 + 5i)(12 - 5i). This is a difference of squares, right? So it's 12² - (5i)². Let me compute that:

12² is 144. (5i)² is 25i², and since i² = -1, that becomes 25*(-1) = -25. Therefore, the denominator is 144 - (-25) = 144 + 25 = 169. Okay, so the denominator simplifies to 169. That's nice.

Now the numerator: (4 + 3i)(12 - 5i). Let me use the distributive property (FOIL method) to multiply these two complex numbers.

First, multiply 4 by 12: 4*12 = 48.

Then, 4*(-5i) = -20i.

Next, 3i*12 = 36i.

Finally, 3i*(-5i) = -15i².

So adding all these terms together: 48 - 20i + 36i -15i².

Combine like terms: the real parts are 48 and -15i², and the imaginary parts are -20i + 36i.

First, let me handle the imaginary parts: -20i + 36i = 16i.

Now, the real parts: 48 -15i². Remember that i² = -1, so -15i² is -15*(-1) = 15. Therefore, the real part is 48 + 15 = 63.

So altogether, the numerator simplifies to 63 + 16i.

Therefore, w = (63 + 16i)/169. Which can be written as (63/169) + (16/169)i.

Now, to find |w|, which is the modulus of w. Since w is a complex number a + bi, where a = 63/169 and b = 16/169, the modulus is sqrt(a² + b²).

So let me compute that:

|w| = sqrt[(63/169)² + (16/169)²] = sqrt[(63² + 16²)/169²] = sqrt(63² + 16²)/sqrt(169²) = sqrt(63² + 16²)/169.

So now I need to compute 63 squared plus 16 squared. Let me calculate each term:

First, 63 squared. Let me recall that 60² = 3600, 63² = (60 + 3)² = 60² + 2*60*3 + 3² = 3600 + 360 + 9 = 3600 + 360 is 3960, plus 9 is 3969.

Then 16 squared is 256. Therefore, adding them together: 3969 + 256. Let me compute that:

3969 + 200 = 4169, then +56 = 4225. Wait, 3969 + 256. Let me check again:

3969 + 256:

Breakdown 256 into 200 + 50 + 6.

3969 + 200 = 4169

4169 + 50 = 4219

4219 + 6 = 4225. Yes, that's correct. So 63² + 16² = 4225.

Therefore, |w| = sqrt(4225)/169. Now sqrt(4225)... Hmm, what is sqrt(4225)?

Well, I know that 65² = 4225 because 60² = 3600, 70² = 4900, so 65² is 4225. Let me check:

65*65: 60*60 + 60*5 + 5*60 + 5*5 = 3600 + 300 + 300 + 25 = 3600 + 600 + 25 = 4225. Yes, correct.

Therefore, sqrt(4225) is 65. Therefore, |w| = 65 / 169.

Now, simplifying 65/169. Let me see if 65 and 169 have a common factor. 65 factors into 5*13. 169 is 13². Therefore, 65 and 169 share a common factor of 13. So divide numerator and denominator by 13:

65 ÷13 = 5, and 169 ÷13 = 13. Therefore, 65/169 simplifies to 5/13.

Therefore, |w| = 5/13.

Wait, let me check that again. 65 divided by 13 is 5, yes. 169 divided by 13 is 13, yes. So 5/13. Therefore, the modulus of w is 5/13.

Let me just recap to make sure I didn't make any mistakes. Starting with z = 1 + i, computed numerator 3z + 1 as 4 + 3i, denominator 5z + 7 as 12 + 5i. Then multiplied numerator and denominator by 12 - 5i, got numerator as 63 + 16i and denominator 169. Then modulus squared was (63² + 16²)/169², which simplified to 4225/169², square root of that is 65/169 = 5/13. Seems correct.

Alternatively, maybe there's a smarter way to compute |w| without computing w explicitly? Let me think. Since |w| = |3z + 1| / |5z + 7|. Because modulus of a quotient is the quotient of the moduli. So maybe I could compute |3z + 1| and |5z + 7| separately and then divide them. Let me try that approach to verify.

First, compute |3z + 1|. Since z = 1 + i, 3z + 1 is 4 + 3i as before. The modulus of 4 + 3i is sqrt(4² + 3²) = sqrt(16 + 9) = sqrt(25) = 5. Okay, that's straightforward.

Then compute |5z + 7|. 5z + 7 is 12 + 5i as before. The modulus is sqrt(12² + 5²) = sqrt(144 + 25) = sqrt(169) = 13. Therefore, |w| = |3z + 1| / |5z + 7| = 5 / 13. Which matches the previous result. So that's a quicker way. So that confirms that the answer is 5/13.

Therefore, regardless of the method, I get 5/13. So I think that's the correct answer.

But just to make sure, let me check my steps again. First method: compute numerator and denominator, multiply by conjugate, get 63 + 16i over 169, then modulus squared is (63² + 16²)/169², which is 4225/28561, square root is 65/169 = 5/13. Second method: modulus of numerator is 5, modulus of denominator is 13, so 5/13. Both methods agree. So I think that's solid.

Alternatively, maybe I can check with another approach. Let me think. Suppose I didn't remember that |a/b| = |a|/|b|. But even so, the first method worked. So I think that's safe.

Alternatively, maybe I can compute w directly as a complex number and then compute modulus. Let me try that again. So w = (4 + 3i)/(12 + 5i). If I write this as (4 + 3i)/(12 + 5i), and then divide them, but maybe I can write it as a decimal or something? Probably not necessary, but just to check.

Alternatively, maybe I can write w in terms of real and imaginary parts. Wait, I already did that. So w = 63/169 + 16i/169. Then modulus squared is (63/169)^2 + (16/169)^2 = (63² + 16²)/169² = 4225/28561. Then sqrt(4225)/sqrt(28561) = 65/169 = 5/13. Yep, same result.

Alternatively, maybe I can check if 63 and 16 have any relation? Well, 63 is 7*9, 16 is 2^4. Not sure if that helps. But anyway, the key was that 63² + 16² = 4225, which is 65². So that's a key step. Let me check that again. 63² is 3969, 16² is 256. 3969 + 256. Let me add them again:

3969 + 256:

Start with 3969 + 200 = 4169

Then add 56: 4169 + 50 = 4219, then +6 = 4225. Correct. And 65² is 4225. So that's correct.

Therefore, all steps check out. Therefore, the modulus of w is 5/13.

I think that's thorough. I considered two different methods, checked my calculations multiple times, and everything seems consistent. So I feel confident that the answer is 5/13.

**Final Answer**
The modulus of $ w $ is \boxed{\dfrac{5}{13}}.

To find the modulus of the complex number $ w = \dfrac{3z + 1}{5z + 7} $, where $ z = 1 + i $, we proceed step by step.

---

### Step 1: Compute the Numerator and Denominator

We begin by substituting $ z = 1 + i $ into the expressions for the numerator and denominator.

- **Numerator**:
  $$
  3z + 1 = 3(1 + i) + 1 = 3 + 3i + 1 = 4 + 3i
  $$

- **Denominator**:
  $$
  5z + 7 = 5(1 + i) + 7 = 5 + 5i + 7 = 12 + 5i
  $$

So,
$$
w = \frac{4 + 3i}{12 + 5i}
$$

---

### Step 2: Simplify the Complex Fraction

To simplify the complex fraction, we multiply the numerator and denominator by the **complex conjugate** of the denominator, which is $ 12 - 5i $:

$$
w = \frac{(4 + 3i)(12 - 5i)}{(12 + 5i)(12 - 5i)}
$$

#### Compute the Denominator:

$$
(12 + 5i)(12 - 5i) = 12^2 - (5i)^2 = 144 - (-25) = 144 + 25 = 169
$$

#### Compute the Numerator:

Use the distributive property (FOIL):

$$
(4 + 3i)(12 - 5i) = 4 \cdot 12 + 4 \cdot (-5i) + 3i \cdot 12 + 3i \cdot (-5i)
$$
$$
= 48 - 20i + 36i - 15i^2 = 48 + 16i - 15(-1) = 48 + 16i + 15 = 63 + 16i
$$

So,
$$
w = \frac{63 + 16i}{169}
$$

---

### Step 3: Compute the Modulus of $ w $

The modulus of a complex number $ a + bi $ is given by:

$$
|w| = \sqrt{a^2 + b^2}
$$

Here, $ a = \frac{63}{169} $, $ b = \frac{16}{169} $, so:

$$
|w| = \sqrt{\left(\frac{63}{169}\right)^2 + \left(\frac{16}{169}\right)^2} = \frac{1}{169} \sqrt{63^2 + 16^2}
$$

Compute the squares:

- $ 63^2 = 3969 $
- $ 16^2 = 256 $

$$
|w| = \frac{1}{169} \sqrt{3969 + 256} = \frac{1}{169} \sqrt{4225}
$$

Now, $ \sqrt{4225} = 65 $, so:

$$
|w| = \frac{65}{169}
$$

Simplify the fraction:

$$
\frac{65}{169} = \frac{5}{13}
$$

---

### Final Answer

$$
\boxed{\dfrac{5}{13}}
$$

Solution: Plugging in, we have $w = \dfrac{3(1+i)+1}{5(1+i)+7} = \dfrac{4+3i}{12+5i}$. We could write this in the form $a+bi$ and take the magnitude, but it's easier to use the fact that, for all complex numbers $a$ and $b$, $\left|\dfrac{a}{b}\right| = \dfrac{|a|}{|b|}$. The magnitude of the numerator is $\sqrt{3^2+4^2} = \sqrt{25} = 5$, and the magnitude of the denominator is $\sqrt{12^2 + 5^2} = \sqrt{169} = 13$. So $|w| = \boxed{\frac{5}{13}}$.