To solve the inequality [tex]\( |2x - 9| \geq 13 \)[/tex], we need to consider the definition of absolute value and split it into two separate inequalities.
The absolute value inequality [tex]\( |2x - 9| \geq 13 \)[/tex] can be rewritten as:
1. [tex]\( 2x - 9 \geq 13 \)[/tex]
2. [tex]\( 2x - 9 \leq -13 \)[/tex]
Let's solve each inequality step by step.
### Solving [tex]\( 2x - 9 \geq 13 \)[/tex]
1. Add 9 to both sides:
[tex]\[
2x - 9 + 9 \geq 13 + 9
\][/tex]
which simplifies to:
[tex]\[
2x \geq 22
\][/tex]
2. Divide both sides by 2:
[tex]\[
\frac{2x}{2} \geq \frac{22}{2}
\][/tex]
which simplifies to:
[tex]\[
x \geq 11
\][/tex]
### Solving [tex]\( 2x - 9 \leq -13 \)[/tex]
1. Add 9 to both sides:
[tex]\[
2x - 9 + 9 \leq -13 + 9
\][/tex]
which simplifies to:
[tex]\[
2x \leq -4
\][/tex]
2. Divide both sides by 2:
[tex]\[
\frac{2x}{2} \leq \frac{-4}{2}
\][/tex]
which simplifies to:
[tex]\[
x \leq -2
\][/tex]
### Combining the Solutions
From the above steps, we have two ranges for [tex]\( x \)[/tex]:
1. [tex]\( x \geq 11 \)[/tex]
2. [tex]\( x \leq -2 \)[/tex]
This means that [tex]\( x \)[/tex] must either be less than or equal to [tex]\(-2\)[/tex] or greater than or equal to [tex]\(11\)[/tex].
Therefore, the solution to the inequality [tex]\( |2x - 9| \geq 13 \)[/tex] is:
[tex]\[
x \leq -2 \text{ or } x \geq 11
\][/tex]
The correct answer is:
B. [tex]\( x \leq -2 \)[/tex] or [tex]\( x \geq 11 \)[/tex]
