Question

solve the inequality for x.
|2x + 9| + 5 < 22

select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution is one or more intervals. the solution is
(simplify your answer. type your answer in interval notation. use integers or fractions for any numbers in the expression.)

b. there are only one or two solutions. the solution set is { }
(type an integer or a fraction. use a comma to separate answers as needed )

c. the solution set is ∅.

Explanation:

Step1: Isolate the absolute value

Subtract 5 from both sides of the inequality \(|2x + 9| + 5 < 22\) to get \(|2x + 9| < 22 - 5\).
\( |2x + 9| < 17 \)

Step2: Solve the compound inequality

The absolute value inequality \(|A| < B\) (where \(B>0\)) is equivalent to \(-B < A < B\). So we have \(-17 < 2x + 9 < 17\).

Step3: Solve the left - hand inequality

Subtract 9 from both sides of \(-17 < 2x + 9\): \(-17-9 < 2x\), which simplifies to \(-26 < 2x\). Then divide both sides by 2: \(-13 < x\).

Step4: Solve the right - hand inequality

Subtract 9 from both sides of \(2x + 9 < 17\): \(2x < 17 - 9\), which simplifies to \(2x < 8\). Then divide both sides by 2: \(x < 4\).

Answer:

A. The solution is one or more intervals. The solution is \((-13,4)\)