Question

solve and graph the solution set on a number line.
$2|2x - 1| + 4 > 14$
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution set is . (type your answer in interval notation.)
b. the solution set is $\emptyset$.

Explanation:

Step1: Isolate absolute value term

Subtract 4 from both sides:
$2|2x-1| + 4 - 4 > 14 - 4$
$2|2x-1| > 10$

Step2: Simplify the inequality

Divide both sides by 2:
$\frac{2|2x-1|}{2} > \frac{10}{2}$
$|2x-1| > 5$

Step3: Split absolute value inequality

For $|A| > B$, solve $A > B$ and $A < -B$:

1. $2x - 1 > 5$
2. $2x - 1 < -5$

Step4: Solve first inequality

Add 1, then divide by 2:
$2x > 5 + 1$
$2x > 6$
$x > 3$

Step5: Solve second inequality

Add 1, then divide by 2:
$2x < -5 + 1$
$2x < -4$
$x < -2$

Answer:

A. The solution set is $(-\infty, -2) \cup (3, \infty)$.

(For the number line graph: Draw an open circle at $x=-2$ with an arrow pointing left, and an open circle at $x=3$ with an arrow pointing right.)