To determine the solution to the inequality [tex]\(|4x - 4| \geq 8\)[/tex], we will break it down step-by-step:
### Step 1: Understanding the Absolute Value Inequality
The given inequality is [tex]\(|4x - 4| \geq 8\)[/tex]. An inequality involving an absolute value, [tex]\(|A| \geq k\)[/tex], where [tex]\(k\)[/tex] is a positive constant, can be split into two inequalities:
[tex]\[ A \leq -k \quad \text{or} \quad A \geq k \][/tex]
### Step 2: Splitting the Inequality
For our inequality [tex]\(|4x - 4| \geq 8\)[/tex], let's set [tex]\(A = 4x - 4\)[/tex] and [tex]\(k = 8\)[/tex]. Therefore, we have two cases:
[tex]\[ 4x - 4 \leq -8 \quad \text{or} \quad 4x - 4 \geq 8 \][/tex]
### Step 3: Solving Each Case
We will solve each inequality separately:
1. First case: [tex]\(4x - 4 \leq -8\)[/tex]
[tex]\[
4x - 4 \leq -8 \\
4x \leq -8 + 4 \\
4x \leq -4 \\
x \leq -1
\][/tex]
2. Second case: [tex]\(4x - 4 \geq 8\)[/tex]
[tex]\[
4x - 4 \geq 8 \\
4x \geq 8 + 4 \\
4x \geq 12 \\
x \geq 3
\][/tex]
### Step 4: Combining the Results
The solutions to the two inequalities are:
- [tex]\(x \leq -1\)[/tex]
- [tex]\(x \geq 3\)[/tex]
Therefore, the solution to the original inequality [tex]\(|4x - 4| \geq 8\)[/tex] is:
[tex]\[ x \leq -1 \quad \text{or} \quad x \geq 3 \][/tex]
### Conclusion
After solving the inequality, we find that the correct solution to the inequality [tex]\(|4x - 4| \geq 8\)[/tex] is:
[tex]\[ x \leq -1 \quad \text{or} \quad x \geq 3 \][/tex]
Hence, the correct answer is:
[tex]\[ x \leq -1 \quad \text{or} \quad x \geq 3 \][/tex]
