https://file.aiquickdraw.com/mathbot/file/1726520228094d7f3jnea.jpg

The equation in the image is:

\[
|4 + 7x| - 15 = k
\]

We are asked to find the value of \(k\) for which this equation has exactly **one real solution**.

### Step-by-Step Solution:

1. **Isolate the absolute value expression:**
   \[
   |4 + 7x| = k + 15
   \]
   This is because adding 15 to both sides simplifies the equation.

2. **Analyze the nature of absolute value:**
   The absolute value function \( |4 + 7x| \) has two cases:
   - **Case 1:** \( 4 + 7x = k + 15 \) (when the expression inside the absolute value is non-negative).
   - **Case 2:** \( 4 + 7x = -(k + 15) \) (when the expression inside the absolute value is negative).

3. **Condition for exactly one solution:**
   For there to be exactly **one real solution**, the expression inside the absolute value must be zero at that value of \(x\), meaning:

   \[
   4 + 7x = 0
   \]

   Solving for \(x\):

   \[
   x = -\frac{4}{7}
   \]

4. **Find \(k\) based on this condition:**
   Substitute \(x = -\frac{4}{7}\) into the original equation:

   \[
   |4 + 7\left(-\frac{4}{7}\right)| - 15 = k
   \]

   Simplifying:

   \[
   |4 - 4| - 15 = k
   \]

   \[
   0 - 15 = k
   \]

   \[
   k = -15
   \]

Thus, the value of \(k\) that gives exactly one real solution is **\(k = -15\)**.

### Conclusion:
For \(k = -15\), the equation has one real solution.

Would you like more details or explanations on any part of the solution? Here are some related questions:

1. What would happen if \(k\) had a different value?
2. Can you explain the two cases in solving absolute value equations?
3. What if the equation had two solutions instead of one?
4. How does the value of \(k\) affect the graph of this equation?
5. Can we solve for \(k\) if we are given a range of solutions?

**Tip:** When solving absolute value equations, always consider both the positive and negative cases, unless constrained by specific conditions.

For which value of k does the equation |4 + 7x| - 15 = k have one real solution? How do you know?

Learn how to solve the absolute value equation |4 + 7x| - 15 = k for a specific value of k where the equation has exactly one real solution. This problem involves absolute value properties and linear equation concepts, suitable for students in Grades 9-12.

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