To solve the absolute value equation \(|5x + 4| = 3x - 16\), we need to consider two cases because the absolute value can be equal to either a positive or negative value:

### Case 1: \(5x + 4 = 3x - 16\)

1. Subtract \(3x\) from both sides:
   \[
   5x - 3x + 4 = -16
   \]
   \[
   2x + 4 = -16
   \]

2. Subtract 4 from both sides:
   \[
   2x = -20
   \]

3. Divide both sides by 2:
   \[
   x = -10
   \]

### Case 2: \(5x + 4 = -(3x - 16)\)

1. Distribute the negative sign on the right side:
   \[
   5x + 4 = -3x + 16
   \]

2. Add \(3x\) to both sides:
   \[
   5x + 3x + 4 = 16
   \]
   \[
   8x + 4 = 16
   \]

3. Subtract 4 from both sides:
   \[
   8x = 12
   \]

4. Divide both sides by 8:
   \[
   x = \frac{12}{8} = \frac{3}{2}
   \]

### Solution Verification:

We need to check if both solutions are valid by substituting them back into the original equation.

1. For \(x = -10\):
   \[
   |5(-10) + 4| = | -50 + 4 | = |-46| = 46
   \]
   \[
   3(-10) - 16 = -30 - 16 = -46
   \]
   Since \(46 = -46\) is not true, \(x = -10\) is **not** a valid solution.

2. For \(x = \frac{3}{2}\):
   \[
   |5 \cdot \frac{3}{2} + 4| = |7.5 + 4| = |11.5| = 11.5
   \]
   \[
   3 \cdot \frac{3}{2} - 16 = 4.5 - 16 = -11.5
   \]
   Since \(11.5 = -11.5\) is not true, \(x = \frac{3}{2}\) is also **not** a valid solution.

### Conclusion:
There are **no valid solutions** for the equation \(|5x + 4| = 3x - 16\).

Would you like more details or have any questions about this process?

### Related Questions:

1. How do you solve absolute value equations when both sides contain variables?
2. What are the properties of absolute values that can affect solutions?
3. How can you graph absolute value equations to understand their solutions visually?
4. How do you handle absolute value inequalities?
5. What are the different types of solutions possible in absolute value equations?

**Tip:** When solving absolute value equations, always verify potential solutions by substituting them back into the original equation.

Explore how to solve the absolute value equation |5x + 4| = 3x - 16 using a step-by-step approach. Learn key algebraic concepts including splitting absolute value equations and checking for extraneous solutions. This solution is designed for students in Grades 9-11.

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