Question

type the correct answer in each box. use numerals instead of words. what is the solution set of this inequality? 2 < |x + 2| < 6

Explanation:

Step1: Split compound inequality

Split into two inequalities:
$|x + 2| > 2$ and $|x + 2| < 8$

Step2: Solve $|x + 2| < 8$

Rewrite without absolute value:
$-8 < x + 2 < 8$
Subtract 2 from all parts:
$-8 - 2 < x < 8 - 2$
$\implies -10 < x < 6$

Step3: Solve $|x + 2| > 2$

Rewrite as two inequalities:
$x + 2 > 2$ or $x + 2 < -2$
Solve first: $x > 2 - 2 \implies x > 0$
Solve second: $x < -2 - 2 \implies x < -4$

Step4: Find intersections

Intersection of $-10 < x < 6$ and $x < -4$: $-10 < x < -4$
Intersection of $-10 < x < 6$ and $x > 0$: $0 < x < 6$
Combine with union: $(-10, -4) \cup (0, 6)$

Answer:

$(-10, -4) \cup (0, 6)$