We are tasked with solving the equation:

\[
5|6 + 3x| - 8 = 52
\]

### Step 1: Isolate the absolute value term

Add 8 to both sides to eliminate the -8 on the left-hand side:

\[
5|6 + 3x| = 52 + 8
\]
\[
5|6 + 3x| = 60
\]

### Step 2: Solve for the absolute value

Now, divide both sides by 5 to isolate the absolute value:

\[
|6 + 3x| = \frac{60}{5}
\]
\[
|6 + 3x| = 12
\]

### Step 3: Remove the absolute value

The absolute value equation \( |6 + 3x| = 12 \) implies two cases:

1. **Case 1**: \( 6 + 3x = 12 \)
2. **Case 2**: \( 6 + 3x = -12 \)

### Case 1: \( 6 + 3x = 12 \)

Subtract 6 from both sides:

\[
3x = 12 - 6
\]
\[
3x = 6
\]

Now, divide by 3:

\[
x = \frac{6}{3} = 2
\]

### Case 2: \( 6 + 3x = -12 \)

Subtract 6 from both sides:

\[
3x = -12 - 6
\]
\[
3x = -18
\]

Now, divide by 3:

\[
x = \frac{-18}{3} = -6
\]

### Final Answer:

The solutions are \( x = 2 \) and \( x = -6 \).

### Verification:
For both \( x = 2 \) and \( x = -6 \), the equation holds true when substituted back into the original equation.

Would you like more details on any step?

Here are 5 related questions to explore:

1. How do you solve equations involving absolute values in general?
2. What happens if the absolute value is negative inside the equation?
3. How do you check the validity of solutions to an equation?
4. What is the difference between absolute value equations and inequalities?
5. How do you graph the solution sets of absolute value equations?

**Tip**: When solving absolute value equations, always remember to consider both the positive and negative scenarios!

Learn how to solve the absolute value equation 5|6 + 3x| - 8 = 52 with step-by-step guidance. This problem focuses on isolating the absolute value, solving linear equations, and verifying solutions. Suitable for students in Grades 8-10.

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