Triangle with sides 15, 25, and x — valid range of x: The three sides of a triangle are 15, 25, and x (in units). Which inequality correctly describes x?
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A10< x < 40
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B10≤ x ≤ 40
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C10 ≤ x < 40
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D10 < x ≤ 40
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ENone of these
Answer
Correct Answer: 10< x < 40
Explanation
Introduction / Context:Using triangle inequality, the unknown side must be strictly less than the sum of the other two and strictly greater than their difference.
Given Data / Assumptions:Known sides are 15 and 25; unknown is x.
Concept / Approach:For sides a, b, x: |a − b| < x < a + b. Substitute a = 15, b = 25.
Step-by-Step Solution:Lower bound: |25 − 15| = 10 → x > 10Upper bound: 25 + 15 = 40 → x < 40Range: 10 < x < 40
Verification / Alternative check:Any endpoint 10 or 40 would make a degenerate triangle (not allowed).
Why Other Options Are Wrong:They include equality or misplace inequality directions.
Common Pitfalls:Allowing equality at the bounds; strict inequalities are required.
Final Answer:10< x < 40