Solve the following absolute value inequality.

[tex]\[
\frac{|2x - 3|}{5} \ \textgreater \ 3
\][/tex]

Find the range for [tex]\( x \)[/tex].



Answer :

Certainly! Let's solve the absolute value inequality step by step.

Given:
[tex]\[ \frac{|2x - 3|}{5} > 3 \][/tex]

First, we will isolate the absolute value expression. To do that, we multiply both sides of the inequality by 5:
[tex]\[ |2x - 3| > 3 \cdot 5 \][/tex]
[tex]\[ |2x - 3| > 15 \][/tex]

Next, recall the definition of absolute value. For an inequality involving an absolute value:
[tex]\[ |A| > B \quad \text{(where B > 0)} \][/tex]
this implies:
[tex]\[ A > B \quad \text{or} \quad A < -B \][/tex]

In context with our problem:
[tex]\[ 2x - 3 > 15 \quad \text{or} \quad 2x - 3 < -15 \][/tex]

We will now solve these two inequalities separately.

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

Add 3 to both sides:
[tex]\[ 2x > 18 \][/tex]

Divide by 2:
[tex]\[ x > 9 \][/tex]

2. Solving [tex]\(2x - 3 < -15\)[/tex]:
[tex]\[ 2x - 3 < -15 \][/tex]

Add 3 to both sides:
[tex]\[ 2x < -12 \][/tex]

Divide by 2:
[tex]\[ x < -6 \][/tex]

Therefore, the solution to the inequality [tex]\(\frac{|2x - 3|}{5} > 3\)[/tex] is:
[tex]\[ x > 9 \quad \text{or} \quad x < -6 \][/tex]

Hence, the complete solution to the given inequality is:
[tex]\[ x > 9 \quad \text{or} \quad x < -6 \][/tex]