Edexcel A-Level Mathematics A2 Fp2 1 Inequalities Questions

Practise solving FP2 inequalities algebraically, including modulus and rational forms where graphical or numerical-only solutions are not accepted.

Exam points

split modulus inequalities into algebraic cases and combine the valid intervals
use critical values from zeros and undefined points to solve rational inequalities

Covered syllabus points

FP2.1.1 - Manipulation and solution

[Maximum number: 6]

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Use algebra to determine the values of x for which

\frac{x+1}{(x-3)(x+2)}\le 1-\frac{2}{x-3}

Critical values from the denominator are x=-2 and x=3.

Equating the two sides:

\frac{x+1}{(x-3)(x+2)}=1-\frac{2}{x-3}

x+1=x^2-x-6-2(x+2)

x^2-4x-11=0.

x=\frac{4\pm\sqrt{16+44}}{2}=2\pm\sqrt{15}

Therefore

x<-2,\qquad 2-\sqrt{15}\le x<3,\qquad x\ge 2+\sqrt{15}

[Maximum number: 6]

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Use algebra to determine the set of values of x for which
$\frac{x^2-9}{|x+8|}>6-2x.$

Critical value:
x=-8
Using |x+8|=x+8 gives critical values
$x=3,\\quad x=-\frac{19}{3}.$
Using |x+8|=-(x+8) gives critical values
$x=3,\quad x=-13.$ $x=-13,\quad x=-\frac{19}{3},\quad x=3$ $-13<x<-8,\quad -8<x<-\frac{19}{3},\quad x>3$
Allow equivalent interval notation. Note that $-13<x<-\frac{19}{3},\ x\ne -8$ would count as two correct inequalities; $-13<x<-\frac{19}{3},\ x>3$ is also accepted as two correct inequalities.

[Maximum number: 8]

Use algebra to determine the set of values of x for which

\frac{x}{2-x} \leqslant \frac{x+3}{x}

(Solutions relying entirely on graphical methods are not acceptable.)

$\frac{x}{2-x},\ \frac{x+3}{x}$

Way 1
$\frac{x}{2-x}-\frac{x+3}{x},\ 0$ $\frac{x}{2-x}-\frac{x+3}{x} \Rightarrow \frac{x^2-(2-x)(x+3)}{x(2-x)},\ 0$ x=0, 2 $x^2-(2-x)(x+3)=0 \Rightarrow 2x^2+x-6=0$ $x=\frac32,\ -2$ $x<-2,\quad 0<x<\frac32,\quad x>2$
(8)

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Covered syllabus points

Question 1

Question 2

Question 3