To solve the inequality [tex]\( |x - 2| \leq 9 \)[/tex], we need to break it down into two separate inequalities based on the definition of absolute value. The absolute value inequality [tex]\( |x - 2| \leq 9 \)[/tex] translates to:
[tex]\[
-9 \leq x - 2 \leq 9
\][/tex]
This compound inequality can be split into two simpler inequalities:
1. [tex]\( x - 2 \leq 9 \)[/tex]
2. [tex]\( x - 2 \geq -9 \)[/tex]
Let's solve each of these inequalities separately.
### Solving [tex]\( x - 2 \leq 9 \)[/tex]:
Add 2 to both sides of the inequality:
[tex]\[
x - 2 + 2 \leq 9 + 2
\][/tex]
Simplifying this gives:
[tex]\[
x \leq 11
\][/tex]
### Solving [tex]\( x - 2 \geq -9 \)[/tex]:
Add 2 to both sides of the inequality:
[tex]\[
x - 2 + 2 \geq -9 + 2
\][/tex]
Simplifying this gives:
[tex]\[
x \geq -7
\][/tex]
### Combining the Inequalities:
Now we combine the results of the two inequalities:
[tex]\[
-7 \leq x \leq 11
\][/tex]
Hence, the solution to the inequality [tex]\( |x - 2| \leq 9 \)[/tex] is:
[tex]\[
-7 \leq x \leq 11
\][/tex]
So, the correct answer is:
b. [tex]\( -7 \leq x \leq 11 \)[/tex]
