To solve the inequality [tex]\( |2x - 4| \geq 6 \)[/tex], we'll need to consider the two scenarios where the absolute value expression [tex]\( |2x - 4| \)[/tex] can be either positive or negative. We'll break the absolute value inequality into two separate inequalities and solve each one.
### Step-by-Step Solution
1. Break down the absolute value inequality:
The absolute value inequality [tex]\( |2x - 4| \geq 6 \)[/tex] translates to the following two cases:
- [tex]\( 2x - 4 \geq 6 \)[/tex]
- [tex]\( 2x - 4 \leq -6 \)[/tex]
2. Solve the first inequality [tex]\( 2x - 4 \geq 6 \)[/tex]:
- Add 4 to both sides: [tex]\( 2x \geq 10 \)[/tex]
- Divide both sides by 2: [tex]\( x \geq 5 \)[/tex]
3. Solve the second inequality [tex]\( 2x - 4 \leq -6 \)[/tex]:
- Add 4 to both sides: [tex]\( 2x \leq -2 \)[/tex]
- Divide both sides by 2: [tex]\( x \leq -1 \)[/tex]
4. Combine the solutions:
The solutions from both inequalities are combined using "or" because the absolute value inequality indicates that either condition can be true. Thus, we have:
- [tex]\( x \leq -1 \)[/tex] or [tex]\( x \geq 5 \)[/tex]
5. Select the correct answer from the given choices:
Review the options given:
- a. [tex]\( x \leq -1 \)[/tex] or [tex]\( x \geq 1 \)[/tex]
- b. [tex]\( x \leq -5 \)[/tex] or [tex]\( x \geq 5 \)[/tex]
- c. [tex]\( x \leq 5 \)[/tex] or [tex]\( x \geq 1 \)[/tex]
- d. [tex]\( x \leq -1 \)[/tex] or [tex]\( x \geq 5 \)[/tex]
The correct solution matches the inequality [tex]\( x \leq -1 \)[/tex] or [tex]\( x \geq 5 \)[/tex].
Given this, the correct choice is:
d. [tex]\( x \leq -1 \)[/tex] or [tex]\( x \geq 5 \)[/tex]
