Question

what is the solution to the inequality $-6 + |2p + 3| > 7$?
○ $p < -8$ or $p > 5$
○ $-8 < p < 5$
○ $p < -2$ or $p > -1$
○ $-2 > p > -1$

Explanation:

Step1: Isolate the absolute value

Add 6 to both sides:
$$|-6 + |2p+3| + 6 > 7 + 6$$
$$|2p+3| > 13$$

Step2: Split into two inequalities

For absolute value $|A| > B$, solve $A > B$ and $A < -B$:
$$2p+3 > 13 \quad \text{or} \quad 2p+3 < -13$$

Step3: Solve first inequality

Subtract 3, then divide by 2:
$$2p > 13 - 3$$
$$2p > 10$$
$$p > 5$$

Step4: Solve second inequality

Subtract 3, then divide by 2:
$$2p < -13 - 3$$
$$2p < -16$$
$$p < -8$$

Answer:

$p < -8$ or $p > 5$ (corresponding to the first option)