Math Problem Statement
Solution
The problem in the image is asking to prove the following implication:
To prove this statement, we can proceed by manipulating the inequality and analyzing the conditions under which the implication holds.
Step-by-Step Proof:
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Given condition: This means that and are at most 2 units apart in the real number line.
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Target condition: We want to show that under this condition: That is, the value of should be within 2 units of .
Approach:
From the given condition, , we know that:
Add 2 to both sides of this inequality:
This directly shows that , which is exactly what we needed to prove.
Conclusion:
Therefore, the implication holds, and we have proven that:
Let me know if you need further clarification!
Here are 5 related questions:
- How does adding or subtracting constants affect inequalities?
- Can the same approach be used to solve other absolute value inequalities?
- What would happen if we changed the constant on the right-hand side of the inequalities?
- How does this proof generalize to higher dimensions or other types of functions?
- What properties of absolute values allow us to handle inequalities in this way?
Tip: When dealing with absolute value inequalities, always try to rewrite them as double inequalities to gain more insight into the solution set.
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Math Problem Analysis
Mathematical Concepts
Real Analysis
Inequalities
Absolute Value
Formulas
|x - y| <= 2
|x + 2| <= 2
Theorems
Properties of Absolute Values
Implication in Inequalities
Suitable Grade Level
Undergraduate Mathematics
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