Solve |5x + 3| ≥ 2x + 7: Absolute Value Inequality Analysis

Consider the inequality: $\left|5x + 3\right| \geq 2x + 7$.

This inequality divides into two cases:

(1) $5x + 3 \geq 2x + 7$ and (2) $5x + 3 \leq -(2x + 7)$.

For inequality (1):

$5x + 3 \geq 2x + 7$

Subtract $2x$ from both sides:

$3x + 3 \geq 7$

Subtract 3 from both sides:

$3x \geq 4$

Divide both sides by 3:

$x \geq \frac{4}{3}$

For inequality (2):

$5x + 3 \leq -2x - 7$

Add $2x$ to both sides:

$7x + 3 \leq -7$

Subtract 3 from both sides:

$7x \leq -10$

Divide both sides by 7:

$x \leq -\frac{10}{7}$

The solution sets are $x \geq \frac{4}{3}$ and $x \leq -\frac{10}{7}$.