Solve |4x - 5| ≤ 2x + 9: Absolute Value Inequality Challenge

Let's solve the inequality: $\left|4x - 5\right| \leq 2x + 9$.

This splits into two cases:

(1) $4x - 5 \leq 2x + 9$ and (2) $4x - 5 \geq -(2x + 9)$.

For inequality (1):

$4x - 5 \leq 2x + 9$

Subtract $2x$ from both sides:

$2x - 5 \leq 9$

Add 5 to both sides:

$2x \leq 14$

Divide both sides by 2:

$x \leq 7$

For inequality (2):

$4x - 5 \geq -2x - 9$

Add $2x$ to both sides:

$6x - 5 \geq -9$

Add 5 to both sides:

$6x \geq -4$

Divide both sides by 6:

$x \geq -\frac{2}{3}$

Combining both solutions gives us $-\frac{2}{3} \leq x \leq 7$.