To solve the absolute value inequality [tex]\(|2x + 5| \geq 15\)[/tex], we need to break this down into two separate inequalities, since absolute value inequalities [tex]\(|A| \geq B\)[/tex] mean:
[tex]\[ A \geq B \quad \text{or} \quad A \leq -B \][/tex]
Let’s break it down step-by-step:
1. First Inequality:
[tex]\[
2x + 5 \geq 15
\][/tex]
Subtract 5 from both sides:
[tex]\[
2x \geq 10
\][/tex]
Divide both sides by 2:
[tex]\[
x \geq 5
\][/tex]
2. Second Inequality:
[tex]\[
2x + 5 \leq -15
\][/tex]
Subtract 5 from both sides:
[tex]\[
2x \leq -20
\][/tex]
Divide both sides by 2:
[tex]\[
x \leq -10
\][/tex]
Now, we combine the solutions from both inequalities. The solution to the absolute value inequality [tex]\(|2x + 5| \geq 15\)[/tex] is:
[tex]\[ x \geq 5 \quad \text{or} \quad x \leq -10 \][/tex]
Given the choices:
a. [tex]\(-\frac{5}{2} \leq x \leq 20\)[/tex]
b. [tex]\(-10 < x < 5\)[/tex]
c. [tex]\(x \geq 5 \quad \text{or} \quad x \leq -10\)[/tex]
d. [tex]\(x \geq -5 \quad \text{or} \quad x \leq 10\)[/tex]
The correct answer is:
c. [tex]\(x \geq 5 \quad \text{or} \quad x \leq -10\)[/tex]
