If x ≤ -2 or x ≥ 8, then what is the appropriate response for |x - 3|?
Understand the Problem
The question is asking to determine the appropriate inequalities regarding the absolute value expression |x - 3|, given the conditions for x. Specifically, we need to find out how the values of x affect the relationship expressed by the absolute value.
Answer
The answer is $|x - 3| \geq 5$.
Answer for screen readers
The appropriate response is $|x - 3| \geq 5$.
Steps to Solve
- Understanding the absolute value inequality
We need to analyze the expression $|x - 3|$. The absolute value represents the distance from the number 3 on the number line. Therefore, we can break it down into two cases.
- Finding the values of $x$
Given the conditions $x \leq -2$ or $x \geq 8$, we'll consider each case separately:
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If $x \leq -2$, substituting $-2$ allows us to find $|x - 3|$: $$ |-2 - 3| = | -5 | = 5 $$
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For values less than $-2$, it can be shown that $|x - 3| > 5$ since these values will move further from 3.
- Evaluating when $x \geq 8$
Now let's evaluate for $x \geq 8$; substituting 8 gives us: $$ |8 - 3| = |5| = 5 $$
For values greater than or equal to 8, it can be concluded that: $$ |x - 3| \geq 5 $$
- Formulating the results
Putting it all together, for all $x \leq -2$ and $x \geq 8$, we find that: $$ |x - 3| \geq 5 $$
This means that, based on the given conditions, the appropriate response is $|x - 3| \geq 5$.
The appropriate response is $|x - 3| \geq 5$.
More Information
For any value of $x$ either less than or equal to $-2$, or greater than or equal to $8$, the distance from 3 will always be 5 or more, due to the properties of absolute value.
Tips
- A common mistake is misinterpreting the absolute value as a strict inequality ($<5$) without considering the ranges where $x$ lies. It's important to note that in both cases ($x \leq -2$ and $x \geq 8$), the absolute difference is equal to or greater than 5, not less.
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