To solve the inequality [tex]\( |4x - 4| \geq 8 \)[/tex], we must consider the definition of the absolute value function. The absolute value inequality [tex]\( |A| \geq B \)[/tex] implies that [tex]\( A \geq B \)[/tex] or [tex]\( A \leq -B \)[/tex]. We will apply this rule to our given inequality.
First, set up the two separate inequalities:
[tex]\[ 4x - 4 \geq 8 \][/tex]
[tex]\[ 4x - 4 \leq -8 \][/tex]
We will solve each of these inequalities separately:
1. Solving [tex]\( 4x - 4 \geq 8 \)[/tex]:
[tex]\[
4x - 4 \geq 8
\][/tex]
Add 4 to both sides:
[tex]\[
4x \geq 12
\][/tex]
Divide both sides by 4:
[tex]\[
x \geq 3
\][/tex]
2. Solving [tex]\( 4x - 4 \leq -8 \)[/tex]:
[tex]\[
4x - 4 \leq -8
\][/tex]
Add 4 to both sides:
[tex]\[
4x \leq -4
\][/tex]
Divide both sides by 4:
[tex]\[
x \leq -1
\][/tex]
Thus, the solution to the inequality [tex]\( |4x - 4| \geq 8 \)[/tex] is:
[tex]\[
x \leq -1 \quad \text{or} \quad x \geq 3
\][/tex]
Therefore, the correct answer is:
[tex]\(\boxed{x \leq -1 \text{ or } x \geq 3}\)[/tex]
