Solve |2x + 3| ≤ 4x - 2: Absolute Value Inequality Analysis

To solve the inequality $\left|2x + 3\right| \leq 4x - 2$, we can consider two cases for the absolute value expression.

Case 1: $2x + 3 \leq 4x - 2$

Rearrange terms:

$3 + 2 \leq 4x - 2x$

$5 \leq 2x$

$x \geq \frac{5}{2}$

Case 2: $-(2x + 3) \leq 4x - 2$

This simplifies to:

$-2x - 3 \leq 4x - 2$

Rearrange terms:

$-3 + 2 \leq 4x + 2x$

$-1 \leq 6x$

$x \geq \frac{-1}{6}$

Combining both conditions, the necessary true statement is:

$x \geq \frac{5}{2}$