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]
