Compare x and y (determine a single relation): I. x^2 − 6x = 7 II. 2y^2 + 13y + 15 = 0 Choose the correct relationship between x and y: x y, x

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Solve this multiple-choice question and choose the correct option.

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Introduction / Context:We must compare any root from each equation. Solving both gives finite root sets; comparing their ranges yields the universal relation if one exists.Given Data / Assumptions: I: x^2 − 6x − 7 = 0 (rearranged). II: 2y^2 + 13y + 15 = 0. Concept / Approach:Solve both quadratics. If the smallest x exceeds the largest y, then x > y for all pairings.Step-by-Step Solution: I: x^2 − 6x − 7 = 0 ⇒ Δ = 36 + 28 = 64 ⇒ x = [6 ± 8]/2 ⇒ x ∈ {7, −1}.II: Δ = 13^2 − 4*2*15 = 169 − 120 = 49 ⇒ √Δ = 7.y = [−13 ± 7]/(4) ⇒ y ∈ { (−6/4)=−1.5 , (−20/4)=−5 }.Hence x ∈ {−1, 7} and y ∈ {−1.5, −5}. The smallest x is −1 and the largest y is −1.5. Since −1 > −1.5 and 7 > any y, every x is greater than every y. Verification / Alternative check:Check all four pairings; each gives x − y > 0. Why Other Options Are Wrong: x < y or x = y: Contradicted by direct computation. Relationship cannot be determined: Incorrect; ordering is strictly separated. Common Pitfalls:Missing the sign when taking square roots or forgetting to rearrange to standard form before solving. Final Answer: x > y

Compare x and y (determine a single relation): I. x^2 − 6x = 7 II. 2y^2 + 13y + 15 = 0 Choose the correct relationship between x and y: x > y, x < y, x = y, or cannot be determined.

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