Solve the Absolute Value Inequality: |5x - 2| < 3x + 8

We start with the inequality: \left|5x - 2\right| < 3x + 8

This absolute value inequality breaks into two separate inequalities:

• 5x - 2 < 3x + 8

• 5x - 2 > -(3x + 8)

Solving the first inequality: 5x - 2 < 3x + 8

• Subtract $3x$ from both sides: 2x - 2 < 8

• Add $2$ to both sides: 2x < 10

• Divide by $2$: x < 5

Solving the second inequality: 5x - 2 > -(3x + 8)

• Distribute the negative: 5x - 2 > -3x - 8

• Add $3x$ to both sides: 8x - 2 > -8

• Add $2$ to both sides: 8x > -6

• Divide by $8$: x > -\frac{3}{4}

The solution is the intersection: x < 5 and x > -\frac{3}{4} . Hence, -\frac{3}{4} < x < 5 .