Questions
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FNDN 2025 O1 Challenge Week 2 Examinations

Single choice

The graphs of the line [math: y=9] and the curve [math: y=x2+2x+1x−1]y=\dfrac{x^2+2x+1}{x-1} intersect at the points [math: (2,9)](2, 9) and [math: (5,9)](5, 9) , as shown in the diagram below. State the solution to the inequality [math: x2+2x+1x−1≤9]\dfrac{x^2+2x+1}{x-1}\leq 9 .

Options
A.[math: 2≤x≤5]2 \leq x \leq 5
B.[math: 1<x≤2 or x≥5]1\lt x \leq 2 \text{ or } x\geq 5
C.[math: x<1]x \lt 1
D.[math: x<1 or 2≤x≤5]x\lt 1 \text{ or } 2 \leq x\leq 5
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Step-by-Step Analysis
We begin by translating the given inequality into a rational inequality that we can analyze piece by piece. Option analysis often hinges on identifying critical points where the expression changes sign or becomes undefined. First, rewrite the expression: y = (x^2 + 2x + 1)/(x - 1) ≤ 9. Notice the numerator factors as (x + 1)^2, so we have (x + 1)^2/(x - 1) ≤ 9. To bring everything to one side, subtract 9 from both sides: (x + 1)^2/(x - 1) - 9 ≤ 0. Get a common denominator to combine: [(x + 1)^2 - 9(x - 1)]/(x - 1) ≤ 0. Expanding the numerator yields (x^2 + 2x + 1) - 9x + 9 = x^2 - 7x + 10,......Login to view full explanation

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