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Worked solution

Use a positive numerator to simplify a cubic ratio

Without using a calculator, solve the inequality x3+3x2+2x+1x3−2x2≥1\dfrac{x^3+3x^2+2x+1}{x^3-2x^2}\geq1. marks

[2016/NYJC/Promo/2]

CJC · Review Problems · Question 5

Answer

x>2x>2.

Solution

Subtract 1: 5x2+2x+1x2(x−2)≥0\dfrac{5x^2+2x+1}{x^2(x-2)}\geq0. Since 5x2+2x+1=5(x+15)2+45>05x^2+2x+1=5(x+\frac15)^2+\frac45>0 for all real xx, only the denominator controls the sign. Away from its poles, x2>0x^2>0. The signs on (−∞,0),(0,2),(2,∞)(-\infty,0),(0,2),(2,\infty) are −,−,+-,-,+. There are no numerator zeros to include, and x=0,2x=0,2 are undefined. Hence x>2x>2.

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The positive numerator divided by x²(x−2) has signs negative, negative, positive across the excluded poles 0 and 2. Only x>2 is a solution.
The positive numerator divided by x²(x−2) has signs negative, negative, positive across the excluded poles 0 and 2. Only x>2 is a solution.
The positive numerator divided by x²(x−2) has signs negative, negative, positive across the excluded poles 0 and 2. Only x>2 is a solution.

Common mistakes

A non-strict inequality need not include any boundary when every critical value is a pole.

Solution notes

The sign chart is independently drawn from the printed solution. Spacing is illustrative; labelled critical values and open/closed endpoints determine the set.

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