To solve the inequality
[tex]\[
8\left(\frac{1}{2} x - \frac{1}{4}\right) > 12 - 2x
\][/tex]
we will proceed step-by-step:
1. Distribute the 8 inside the parentheses:
[tex]\[
8 \left(\frac{1}{2} x - \frac{1}{4}\right) = 8 \cdot \frac{1}{2} x - 8 \cdot \frac{1}{4}
\][/tex]
[tex]\[
= 4x - 2
\][/tex]
2. Rewrite the inequality:
[tex]\[
4x - 2 > 12 - 2x
\][/tex]
3. Add [tex]\(2x\)[/tex] to both sides to get all [tex]\(x\)[/tex]-terms on one side:
[tex]\[
4x - 2 + 2x > 12 - 2x + 2x
\][/tex]
[tex]\[
6x - 2 > 12
\][/tex]
4. Add 2 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
6x - 2 + 2 > 12 + 2
\][/tex]
[tex]\[
6x > 14
\][/tex]
5. Divide both sides by 6 to solve for [tex]\(x\)[/tex]:
[tex]\[
x > \frac{14}{6}
\][/tex]
6. Simplify the fraction:
[tex]\[
x > \frac{14}{6} = \frac{7}{3}
\][/tex]
The solution to the inequality is:
[tex]\[
x > \frac{7}{3}
\][/tex]
Hence, the correct answer is:
A. [tex]\(x > \frac{7}{3}\)[/tex]
