Sure, let's solve the inequality step by step.
Given inequality:
[tex]\[ 8\left(\frac{1}{2} x - \frac{1}{4}\right) > 12 - 2x \][/tex]
Step 1: Distribute the 8 on the left-hand side.
[tex]\[ 8 \cdot \frac{1}{2} x - 8 \cdot \frac{1}{4} > 12 - 2x \][/tex]
[tex]\[ 4x - 2 > 12 - 2x \][/tex]
Step 2: Combine like terms by adding [tex]\(2x\)[/tex] to both sides.
[tex]\[ 4x + 2x - 2 > 12 - 2x + 2x \][/tex]
[tex]\[ 6x - 2 > 12 \][/tex]
Step 3: Isolate the term with [tex]\(x\)[/tex] by adding 2 to both sides.
[tex]\[ 6x - 2 + 2 > 12 + 2 \][/tex]
[tex]\[ 6x > 14 \][/tex]
Step 4: Solve for [tex]\(x\)[/tex] by dividing by 6.
[tex]\[ x > \frac{14}{6} \][/tex]
[tex]\[ x > \frac{7}{3} \][/tex]
Thus, the solution to the inequality is:
[tex]\[ x > \frac{7}{3} \][/tex]
Checking the possible answer choices:
A. [tex]\( x > -5 \)[/tex] is incorrect.
B. [tex]\( x > 7 \)[/tex] is also incorrect.
C. [tex]\( x > \frac{5}{3} \)[/tex] is too lenient, thus incorrect.
D. [tex]\( x > \frac{7}{3} \)[/tex] is correct.
Therefore, the correct answer is:
[tex]\[ \boxed{x > \frac{7}{3}} \][/tex]
