Question

solve the inequality. then graph the solution set and write it in interval - notation. |x - 5| - 7 ≤ - 2
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 7 to both sides of the inequality \(|x - 5|-7\leq - 2\).
\(|x - 5|\leq - 2 + 7\), so \(|x - 5|\leq5\).

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

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

Step3: Solve the compound inequality

Add 5 to all parts of the compound inequality: \(-5+5\leq x-5 + 5\leq5 + 5\).
This simplifies to \(0\leq x\leq10\).

Answer:

A. The solution set is \([0,10]\)
The correct graph is a number - line with a closed circle at 0 and a closed circle at 10, and the line segment between them is shaded. (Corresponds to a graph where the endpoints are filled - in dots and the line between them is shaded, which is not explicitly labeled in the given options but is the correct graphical representation for the interval \([0,10]\)).