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Which of the following is the solution to the inequality [tex]|2x - 3| \ < \ 15[/tex]?

A. [tex]-6 \ < \ x \ < \ 9[/tex]
B. [tex]6 \ < \ x \ < \ 6[/tex]
C. [tex]-9 \ < \ x \ < \ 9[/tex]
D. [tex]x \ < \ -6[/tex] or [tex]x \ > \ 9[/tex]

Answer :

To solve the inequality [tex]\( |2x - 3| < 15 \)[/tex], we need to break it down into two separate inequalities because the absolute value function creates two scenarios.

### Step 1: Remove the Absolute Value
The inequality [tex]\( |2x - 3| < 15 \)[/tex] means that the expression inside the absolute value, [tex]\( 2x - 3 \)[/tex], is less than 15 and greater than -15 simultaneously. So we can write two inequalities:

1. [tex]\( 2x - 3 < 15 \)[/tex]
2. [tex]\( 2x - 3 > -15 \)[/tex]

### Step 2: Solve Each Inequality Separately

#### Inequality 1: [tex]\( 2x - 3 < 15 \)[/tex]

1. Add 3 to both sides:
[tex]\[
2x - 3 + 3 < 15 + 3
\][/tex]
[tex]\[
2x < 18
\][/tex]

2. Divide both sides by 2:
[tex]\[
\frac{2x}{2} < \frac{18}{2}
\][/tex]
[tex]\[
x < 9
\][/tex]

#### Inequality 2: [tex]\( 2x - 3 > -15 \)[/tex]

1. Add 3 to both sides:
[tex]\[
2x - 3 + 3 > -15 + 3
\][/tex]
[tex]\[
2x > -12
\][/tex]

2. Divide both sides by 2:
[tex]\[
\frac{2x}{2} > \frac{-12}{2}
\][/tex]
[tex]\[
x > -6
\][/tex]

### Step 3: Combine the Solutions
Now we combine the two solutions:
[tex]\[
-6 < x < 9
\][/tex]

Thus, the solution to the inequality [tex]\( |2x - 3| < 15 \)[/tex] is:
[tex]\[
-6 < x < 9
\][/tex]

Therefore, the correct answer is:
[tex]\[
-6 < x < 9
\][/tex]

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