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, $-\frac{3}{4} < x < 5$.