To solve the absolute value inequality [tex]\(|2x - 3| \leq 9\)[/tex], we need to understand the properties of absolute values. Specifically, the expression [tex]\(|A| \leq B\)[/tex] implies that [tex]\(-B \leq A \leq B\)[/tex].
Let's apply this property step-by-step.
1. Express the Inequality Without Absolute Value:
[tex]\[
|2x - 3| \leq 9 \implies -9 \leq 2x - 3 \leq 9
\][/tex]
2. Break it into Two Separate Inequalities:
We now have two inequalities:
[tex]\[
-9 \leq 2x - 3 \quad \text{and} \quad 2x - 3 \leq 9
\][/tex]
3. Solve Each Inequality for [tex]\(x\)[/tex]:
- For [tex]\(-9 \leq 2x - 3\)[/tex]:
[tex]\[
-9 \leq 2x - 3
\][/tex]
Add 3 to both sides:
[tex]\[
-9 + 3 \leq 2x
\][/tex]
Simplify:
[tex]\[
-6 \leq 2x
\][/tex]
Divide both sides by 2:
[tex]\[
-3 \leq x
\][/tex]
- For [tex]\(2x - 3 \leq 9\)[/tex]:
[tex]\[
2x - 3 \leq 9
\][/tex]
Add 3 to both sides:
[tex]\[
2x \leq 9 + 3
\][/tex]
Simplify:
[tex]\[
2x \leq 12
\][/tex]
Divide both sides by 2:
[tex]\[
x \leq 6
\][/tex]
4. Combine the Results:
Combining the inequalities, we get:
[tex]\[
-3 \leq x \leq 6
\][/tex]
Thus, the solution set for the inequality [tex]\(|2x - 3| \leq 9\)[/tex] is [tex]\(-3 \leq x \leq 6\)[/tex].
Among the given options, the correct one is:
b. [tex]\(-3 \leq x \leq 6\)[/tex]
