To solve the inequality [tex]\( |2x-3| > 5 \)[/tex], we will start by considering the definition of absolute value inequalities. The absolute value inequality [tex]\( |A| > B \)[/tex] can be split into two separate inequalities: [tex]\( A > B \)[/tex] or [tex]\( A < -B \)[/tex].
In this case, we have:
[tex]\[ |2x - 3| > 5 \][/tex]
We can split this into two separate inequalities:
[tex]\[ 2x - 3 > 5 \quad \text{or} \quad 2x - 3 < -5 \][/tex]
Let's solve each inequality separately.
First Inequality:
[tex]\[ 2x - 3 > 5 \][/tex]
Add 3 to both sides:
[tex]\[ 2x > 8 \][/tex]
Divide both sides by 2:
[tex]\[ x > 4 \][/tex]
Second Inequality:
[tex]\[ 2x - 3 < -5 \][/tex]
Add 3 to both sides:
[tex]\[ 2x < -2 \][/tex]
Divide both sides by 2:
[tex]\[ x < -1 \][/tex]
Therefore, the solution to the inequality [tex]\( |2x - 3| > 5 \)[/tex] is the union of the solutions to these two inequalities.
[tex]\[ x < -1 \quad \text{or} \quad x > 4 \][/tex]
So, the correct answer is:
[tex]\[ x < -1 \quad \text{or} \quad x > 4 \][/tex]
This matches the first option provided:
[tex]\[ \boxed{x < -1 \text{ or } x > 4} \][/tex]
