Question

solve the absolute value inequality.
2|2x - 9| + 12 > 22

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

Explanation:

Step1: Isolate the absolute value

Subtract 12 from both sides of the inequality \(2|2x - 9|+12>22\).
\(2|2x - 9|+12 - 12>22 - 12\)
\(2|2x - 9|>10\)

Step2: Divide by the coefficient of the absolute value

Divide both sides by 2.
\(\frac{2|2x - 9|}{2}>\frac{10}{2}\)
\(|2x - 9|>5\)

Step3: Solve the compound inequality

The absolute value inequality \(|A|>b\) (where \(b>0\)) is equivalent to \(A < - b\) or \(A>b\). So we have two cases:

Case 1: \(2x - 9 < - 5\)

Add 9 to both sides:
\(2x-9 + 9<-5 + 9\)
\(2x<4\)
Divide both sides by 2:
\(x < 2\)

Case 2: \(2x - 9>5\)

Add 9 to both sides:
\(2x-9 + 9>5 + 9\)
\(2x>14\)
Divide both sides by 2:
\(x>7\)

Answer:

A. The solution set in interval notation is \((-\infty, 2)\cup(7, \infty)\)