What is the solution to the inequality [tex]|2x - 3| \ \textgreater \ 5[/tex]?

A. [tex]x \ \textless \ -1[/tex] or [tex]x \ \textgreater \ 4[/tex]

B. [tex]x \ \textless \ 0[/tex] or [tex]x \ \textgreater \ 8[/tex]

C. [tex]0 \ \textless \ x \ \textless \ 8[/tex]

D. [tex]-1 \ \textless \ x \ \textless \ 4[/tex]



Answer :

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]