To solve the absolute value inequality [tex]\( 2|2x - 4| > 8 \)[/tex], follow these steps:
1. Isolate the absolute value expression:
[tex]\[
2|2x - 4| > 8
\][/tex]
Divide both sides by 2:
[tex]\[
|2x - 4| > 4
\][/tex]
2. Interpret the absolute value inequality:
The inequality [tex]\( |2x - 4| > 4 \)[/tex] means that the expression inside the absolute value, [tex]\( 2x - 4 \)[/tex], is either greater than 4 or less than -4.
Therefore, we have two cases to consider:
[tex]\[
2x - 4 > 4 \quad \text{or} \quad 2x - 4 < -4
\][/tex]
3. Solve each case separately:
- Case 1: [tex]\( 2x - 4 > 4 \)[/tex]
[tex]\[
2x - 4 > 4
\][/tex]
Add 4 to both sides:
[tex]\[
2x > 8
\][/tex]
Divide both sides by 2:
[tex]\[
x > 4
\][/tex]
- Case 2: [tex]\( 2x - 4 < -4 \)[/tex]
[tex]\[
2x - 4 < -4
\][/tex]
Add 4 to both sides:
[tex]\[
2x < 0
\][/tex]
Divide both sides by 2:
[tex]\[
x < 0
\][/tex]
4. Combine the results:
The solutions to the inequality [tex]\( 2|2x - 4| > 8 \)[/tex] are [tex]\( x > 4 \)[/tex] and [tex]\( x < 0 \)[/tex].
Thus, the complete solution set is:
[tex]\[
x < 0 \quad \text{or} \quad x > 4
\][/tex]
5. Evaluate the answer choices and select the correct one:
a. [tex]\( -2 < x < 10 \)[/tex]: This is incorrect.
b. [tex]\( x < 0 \)[/tex] or [tex]\( x > 4 \)[/tex]: This matches our solution.
c. [tex]\( x < -2 \)[/tex] or [tex]\( x > 10 \)[/tex]: This is incorrect.
d. [tex]\( 0 < x < 4 \)[/tex]: This is incorrect.
Therefore, the correct answer is:
b. [tex]\( x < 0 \)[/tex] or [tex]\( x > 4 \)[/tex]
