Question

solve the absolute value inequality.
-7|x - 4| ≥ -28

rewrite the inequality without absolute value bars. select the correct choice below and fill in the answer boxes to complete your choice. (simplify your answers.)
a. x - 4 ≥

b. ≤ x - 4 ≤

c. x - 4 ≤ or x - 4 ≥

select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution set in interval notation is . (simplify your answer.)
b. the solution set is ∅.

Explanation:

Step1: Divide both sides by -7

When dividing an inequality by a negative number, the inequality sign flips. So, we divide both sides of \(-7|x - 4| \geq -28\) by \(-7\):
\(\frac{-7|x - 4|}{-7} \leq \frac{-28}{-7}\)
Simplifying gives \(|x - 4| \leq 4\)

Step2: Rewrite absolute value inequality

The absolute value inequality \(|a| \leq b\) (where \(b\geq0\)) is equivalent to \(-b \leq a \leq b\). Here, \(a = x - 4\) and \(b = 4\), so we rewrite \(|x - 4| \leq 4\) as \(-4 \leq x - 4 \leq 4\). So the correct choice for the first part is B with \(-4\) and \(4\).

Step3: Solve the compound inequality

To solve \(-4 \leq x - 4 \leq 4\), we add 4 to all parts:
\(-4 + 4 \leq x - 4 + 4 \leq 4 + 4\)
Simplifying each part: \(0 \leq x \leq 8\)

Step4: Write in interval notation

The interval notation for \(0 \leq x \leq 8\) is \([0, 8]\). So the solution set in interval notation is \([0, 8]\), so the correct choice for the second part is A with \([0, 8]\).

Answer:

First part: B. \(-4 \leq x - 4 \leq 4\)
Second part: A. The solution set in interval notation is \([0, 8]\)