Solve |x-3| ≤ 5: Absolute Value Inequality Step-by-Step

Given:

$|x-3| \leq 5$

Which of the following statements is necessarily true?

To solve the inequality $|x-3| \leq 5$, we need to consider the definition of absolute value inequalities. The inequality $|a| \leq b$ translates to $-b \leq a \leq b$.

Applying this to our expression $|x-3| \leq 5$, we have:

$-5 \leq x-3 \leq 5$.

We add 3 to all parts of the inequality to isolate $x$:

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

This simplifies to $-2 \leq x \leq 8$.