Solve |3x - 4| ≤ 5: Absolute Value Inequality Solution

Given:

$\left|3x - 4\right| \leq 5$

Which of the following statements is necessarily true?

To solve $\left| 3x - 4 \right| \leq 5$, we should consider two scenarios for the absolute value: $3x - 4 \leq 5$ and $3x - 4 \geq -5$.

1. Solving $3x - 4 \leq 5$:

$3x - 4 \leq 5$

Add 4 to both sides:

$3x \leq 9$

Divide both sides by 3:

$x \leq 3$

2. Solving $3x - 4 \geq -5$:

$3x - 4 \geq -5$

Add 4 to both sides:

$3x \geq -1$

Divide both sides by 3:

$x \geq -\frac{1}{3}$

Combining both results, we have $-\frac{1}{3} \leq x \leq 3$, which is the correct answer.