If x ≤ -2 or x ≥ 8, then what can be said about |x - 3|?
Understand the Problem
The question is asking to determine the relationship involving the absolute value of x given the conditions for x. Specifically, it requires analyzing the conditions |x - 3| under the constraints of x being less than or equal to -2 or greater than or equal to 8.
Answer
The answer is \(|x - 3| \geq 5\).
Answer for screen readers
The correct response is (|x - 3| \geq 5).
Steps to Solve
- Identify the constraints on (x)
Given that (x \leq -2) or (x \geq 8), we need to analyze both regions:
- For (x \leq -2)
- For (x \geq 8)
- Evaluate (|x - 3|) for (x \leq -2)
When (x \leq -2), we compute:
$$ |x - 3| = | -2 - 3 | = |-5| = 5 $$
This value increases as (x) becomes more negative, meaning (|x - 3| \geq 5).
- Evaluate (|x - 3|) for (x \geq 8)
When (x \geq 8), we compute:
$$ |x - 3| = |8 - 3| = |5| = 5 $$
Here, (|x - 3|) increases as (x) increases, meaning (|x - 3| \geq 5).
- Combine results from both cases
From both cases, we observe that:
- For (x \leq -2) and (x \geq 8), we found (|x - 3| \geq 5).
- Determine the appropriate response
Based on the evaluations, the statement (|x - 3| \geq 5) is true under the given conditions.
The correct response is (|x - 3| \geq 5).
More Information
Absolute value measures the distance from zero on the number line, and in this context, it reflects how far (x) is away from 3. As derived, both constraints yield distances of at least 5.
Tips
- Misunderstanding absolute value: Failing to account for how absolute values work under different values of (x). Make sure to evaluate both cases carefully to determine the conditions accurately.
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