Problem 125
Question
Prove or disprove: If \(x\) and \(y\) are real numbers with \(y \geq 0\) and \(y(y+1) \leq(x+1)^{2},\) then \(y(y-1) \leq x^{2}\)
Step-by-Step Solution
Verified Answer
The provided statement is disproved.
1Step 1: Rewrite the given conditions
Rewrite \(y(y+1) \leq (x+1)^{2}\) as \[y^2 + y \leq x^{2} + 2x + 1.\] Also, rewrite \(y(y-1) \leq x^{2}\) as \[ y^2 - y \leq x^2.\] Subtracting the second inequality from the first, we get \(2y \leq 2x + 1.\]
2Step 2: Simplify the inequality
Simplify the inequality from Step 1 by dividing both sides by 2 and rearranging terms. This results: \(y \leq x + 0.5\).
3Step 3: Comparison with original statement
This inequality is not the same as the original one provided. So, the given statement is disproved.
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