Inequalities

WFM02Topic 2.12 questions

Algebraic inequalities involving rational functions and the modulus function.

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Original practice questions written in the style of the exam.

Question 16 marks
Use algebra to find the set of values of $x$ for which $\frac{3x}{x - 1} > x + 1$ Give your answer using exact values.
Mark scheme
- M1 Collects on one side over a common denominator: $\dfrac{3x - (x + 1)(x - 1)}{x - 1} > 0$ (1)

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0

A1

\dfrac{-x^2 + 3x + 1}{x - 1} > 0

, or equivalently

\dfrac{x^2 - 3x - 1}{x - 1} < 0

(1)

M1 Finds critical values from

x^2 - 3x - 1 = 0

(1)

A1 Critical values

x = 1

and

x = \dfrac{3 \pm \sqrt{13}}{2}

(1)

M1 Uses a sign diagram or sketch to choose the correct regions (1)

A1

x < \dfrac{3 - \sqrt{13}}{2}

or

1 < x < \dfrac{3 + \sqrt{13}}{2}

(1)