Solve Absolute Value Inequality: |2x + 3| > 4x + 5

We start with the inequality: \left|2x + 3\right| > 4x + 5

This absolute value inequality breaks into two separate inequalities:

• 2x + 3 > 4x + 5

• 2x + 3 < -(4x + 5)

Solving the first inequality: 2x + 3 > 4x + 5

• Subtract $2x$ from both sides: 3 > 2x + 5

• Subtract $5$ from both sides: -2 > 2x

• Divide by $2$: x < -1

Solving the second inequality: 2x + 3 < -(4x + 5)

• Distribute the negative: 2x + 3 < -4x - 5

• Add $4x$ to both sides: 6x + 3 < -5

• Subtract $3$ from both sides: 6x < -8

• Divide by $6$: x < -\frac{4}{3}

The solution is the intersection: x < -1 .