To solve the inequality [tex]\( |2x - 3| > 5 \)[/tex], we need to break it into two separate inequalities based on the properties of the absolute value function.
The absolute value inequality [tex]\( |2x - 3| > 5 \)[/tex] can be split into two cases:
1. [tex]\( 2x - 3 > 5 \)[/tex]
2. [tex]\( 2x - 3 < -5 \)[/tex]
Let's solve these two inequalities one by one.
### Case 1: [tex]\( 2x - 3 > 5 \)[/tex]
1. Start by isolating [tex]\( x \)[/tex].
[tex]\[ 2x - 3 > 5 \][/tex]
2. Add 3 to both sides.
[tex]\[ 2x > 8 \][/tex]
3. Divide both sides by 2.
[tex]\[ x > 4 \][/tex]
So, the solution for the first inequality is [tex]\( x > 4 \)[/tex].
### Case 2: [tex]\( 2x - 3 < -5 \)[/tex]
1. Start by isolating [tex]\( x \)[/tex].
[tex]\[ 2x - 3 < -5 \][/tex]
2. Add 3 to both sides.
[tex]\[ 2x < -2 \][/tex]
3. Divide both sides by 2.
[tex]\[ x < -1 \][/tex]
So, the solution for the second inequality is [tex]\( x < -1 \)[/tex].
### Combining Both Cases
The combined solution to the inequality [tex]\( |2x - 3| > 5 \)[/tex] is:
[tex]\[ x < -1 \quad \text{or} \quad x > 4 \][/tex]
Thus, the correct answer is:
[tex]\[ x < -1 \quad \text{or} \quad x > 4 \][/tex]
Therefore, among the given choices, the correct one is:
[tex]\[ \boxed{x < -1 \quad \text{or} \quad x > 4} \][/tex]
