To solve the equation \( \left| -4 + 5x \right| = 16 \), we need to break it into two cases, since the absolute value expression represents two possible scenarios:

### Case 1: \( -4 + 5x = 16 \)

Solve for \( x \):
\[
-4 + 5x = 16
\]
Add 4 to both sides:
\[
5x = 20
\]
Now, divide by 5:
\[
x = 4
\]

### Case 2: \( -4 + 5x = -16 \)

Solve for \( x \):
\[
-4 + 5x = -16
\]
Add 4 to both sides:
\[
5x = -12
\]
Now, divide by 5:
\[
x = -\frac{12}{5}
\]

### Solutions:
Thus, the two solutions are:
\[
x = 4 \quad \text{or} \quad x = -\frac{12}{5}
\]

Would you like further details or clarification on any step?

Here are five related questions to practice:
1. Solve \( \left| 2x + 3 \right| = 9 \).
2. Solve \( \left| x - 7 \right| = 3 \).
3. Solve \( \left| 4x - 5 \right| = 20 \).
4. Solve \( \left| 2x - 1 \right| = 10 \).
5. Solve \( \left| x + 2 \right| = 6 \).

**Tip**: When solving absolute value equations, always break the equation into two cases: one for the positive scenario and one for the negative scenario.

Learn how to solve the absolute value equation |-4 + 5x| = 16 with a detailed step-by-step solution. This algebraic problem covers concepts like breaking absolute value equations into two cases and solving linear equations. Suitable for students in Grades 8-10.

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