Determine the solution to the inequality. |4x − 4| ≥ 8 x ≤ −1 or x ≥ 3 x ≤ −2 or x ≥ 3 x ≤ −3 or x ≥ 4 x ≤ −4 or x ≥ 4
The Correct Answer and Explanation is:
To solve the inequality (|4x – 4| \geq 8), we will break it down into two separate cases, because the absolute value inequality can be rewritten without the absolute value as two separate inequalities:
- Case 1: (4x – 4 \geq 8)
- Case 2: (4x – 4 \leq -8)
Solving Case 1: (4x – 4 \geq 8)
- Start by isolating (4x):
[
4x – 4 \geq 8
]
[
4x \geq 12
] - Divide both sides by 4:
[
x \geq 3
]
Solving Case 2: (4x – 4 \leq -8)
- Isolate (4x):
[
4x – 4 \leq -8
]
[
4x \leq -4
] - Divide both sides by 4:
[
x \leq -1
]
Combining the Results
From Case 1, we found (x \geq 3) and from Case 2, we found (x \leq -1). Therefore, the complete solution to the inequality (|4x – 4| \geq 8) can be expressed in interval notation as:
[
x \leq -1 \quad \text{or} \quad x \geq 3
]
Conclusion
Thus, the solution is (x \leq -1) or (x \geq 3).
This means that any value of (x) less than or equal to (-1) or greater than or equal to (3) satisfies the original inequality. The critical points that define these intervals are (-1) and (3). Graphically, on the number line, the solution would cover all numbers to the left of (-1) (inclusive) and all numbers to the right of (3) (inclusive).
The provided multiple-choice options were:
- (x \leq -1) or (x \geq 3)
- (x \leq -2) or (x \geq 3)
- (x \leq -3) or (x \geq 4)
- (x \leq -4) or (x \geq 4)
The correct answer is:
(x \leq -1) or (x \geq 3).