To solve the absolute value inequality:
[tex]\[
\frac{|2x + 1|}{3} \geq 5
\][/tex]
Let's start by isolating the absolute value expression. Multiply both sides of the inequality by 3:
[tex]\[
|2x + 1| \geq 15
\][/tex]
The absolute value inequality [tex]\( |2x + 1| \geq 15 \)[/tex] means that the expression inside the absolute value, [tex]\( 2x + 1 \)[/tex], is either greater than or equal to 15, or less than or equal to -15.
This gives us two inequalities to solve:
1. [tex]\( 2x + 1 \geq 15 \)[/tex]
2. [tex]\( 2x + 1 \leq -15 \)[/tex]
### Solving the First Inequality
[tex]\[
2x + 1 \geq 15
\][/tex]
Subtract 1 from both sides:
[tex]\[
2x \geq 14
\][/tex]
Divide both sides by 2:
[tex]\[
x \geq 7
\][/tex]
### Solving the Second Inequality
[tex]\[
2x + 1 \leq -15
\][/tex]
Subtract 1 from both sides:
[tex]\[
2x \leq -16
\][/tex]
Divide both sides by 2:
[tex]\[
x \leq -8
\][/tex]
### Combining the Solutions
The solutions to the original inequality are the values of [tex]\( x \)[/tex] that satisfy either:
[tex]\[
x \geq 7 \quad \text{or} \quad x \leq -8
\][/tex]
Therefore, the final solution is:
[tex]\[
x \geq 7 \quad \text{or} \quad x \leq -8
\][/tex]
