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The given equation is:

\[
-5|3 + 4k| = -115
\]

Let's solve for \( k \) step by step.

### Step 1: Divide both sides by -5 to simplify
\[
|3 + 4k| = \frac{-115}{-5} = 23
\]

### Step 2: Set up two cases for the absolute value

1. \( 3 + 4k = 23 \)
2. \( 3 + 4k = -23 \)

### Case 1: \( 3 + 4k = 23 \)
\[
4k = 23 - 3
\]
\[
4k = 20
\]
\[
k = \frac{20}{4} = 5
\]

### Case 2: \( 3 + 4k = -23 \)
\[
4k = -23 - 3
\]
\[
4k = -26
\]
\[
k = \frac{-26}{4} = -6.5
\]

### Final Answer:
The solutions are:
\[
k = 5 \quad \text{or} \quad k = -6.5
\]

Would you like more details, or do you have any questions?

Here are some related questions you might find interesting:
1. How do you handle equations with absolute values and different coefficients?
2. Can you explain why absolute value equations lead to two cases?
3. What are some methods to verify the solutions of an absolute value equation?
4. How do absolute value equations differ from regular linear equations?
5. Could you solve a system of equations that includes absolute values?

**Tip:** When dealing with absolute value equations, always remember to account for both the positive and negative cases.

Discover how to solve the absolute value equation -5|3 + 4k| = -115 with a clear, step-by-step solution. This problem explores key algebraic concepts, such as absolute value properties and solving linear equations. Ideal for students in Grades 8-10.

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