Question

solve the inequality. then graph the solution set and write it in interval notation.
|x - 2| - 9 ≤ - 3
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution set is
(simplify your answer. type your answer in interval notation. use integers or fractions for any numbers in the expression.)
b. the solution set is one or two points. the solution set is
(type an integer or a fraction. use a comma to separate answers as needed.)
c. the solution set is ∅
choose the correct graph below.

Explanation:

Step1: Isolate the absolute - value expression

Add 8 to both sides of the inequality \(|x - 2|-8\leq - 3\).
\(|x - 2|\leq - 3 + 8\), so \(|x - 2|\leq5\).

Step2: Rewrite the absolute - value inequality as a compound inequality

If \(|u|\leq a\) (\(a\geq0\)), then \(-a\leq u\leq a\). Here \(u=x - 2\) and \(a = 5\), so \(-5\leq x-2\leq5\).

Step3: Solve the compound inequality

Add 2 to all parts of the compound inequality: \(-5+2\leq x-2 + 2\leq5+2\).
We get \(-3\leq x\leq7\).

Answer:

A. The solution set is \([-3,7]\)
The correct graph is D (a number line with a closed - circle at \(-3\), a closed - circle at 7, and the line segment between them shaded).