Why not divide an inequality by x?

x may be negative (the inequality reverses) or zero; you may also lose solutions.

What are critical values?

The roots of the polynomial, which split the number line into intervals of constant sign.

Never negative: zero at x=a, positive elsewhere, so it does not change the sign.

When may you square both sides of an inequality?

When both sides are non-negative.

When is |A|\ge B automatically true?

When B\le0 (where A is defined).

(B-A)(B+A) (difference of two squares).

How do you check a final answer to an inequality?

Test a value from each interval, and each critical value, in the original inequality.

Inequalities with polynomials and rational expressionsAQA A-Level Further Maths: Flashcards

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First step in solving a polynomial inequality?

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First step in solving a polynomial inequality?: Move all terms to one side and factorise.
Why not divide an inequality by $x$ ?: $x$ may be negative (the inequality reverses) or zero; you may also lose solutions.
What are critical values?: The roots of the polynomial, which split the number line into intervals of constant sign.
Sign of $(x-a)^2$ ?: Never negative: zero at $x=a$ , positive elsewhere, so it does not change the sign.
Solve $(x-1)^2(x+1)>0$ .: $x>-1$ , $x\ne1$
Solve $(x-1)(x-2)(x+3)>0$ .: $-3<x<1$ or $x>2$
Solve $(x^2-1)(x^2-4)<0$ .: $-2<x<-1$ or $1<x<2$
When may you square both sides of an inequality?: When both sides are non-negative.
$|A|<B$ is only possible if...?: $B>0$
$|A|<B$ is equivalent to?: $-B<A<B$ (for $B>0$ ), or $A^2<B^2$ .
When is $|A|\ge B$ automatically true?: When $B\le0$ (where $A$ is defined).
Factorise $B^2-A^2$ .: $(B-A)(B+A)$ (difference of two squares).
How do you check a final answer to an inequality?: Test a value from each interval, and each critical value, in the original inequality.

Exam questions on Inequalities with polynomials and rational expressions

Let $p(x)=x^3-7x+6$ .
Solve $x^3+6<7x$ .2 marks
Let $q(x)=x^4-5x^2+4$ .
Find the set of values of $x$ for which $\sqrt{q(x)}$ is real.2 marks
A cubic curve $C_1$ has equation $y=x^3-x$ and a parabola $C_2$ has equation $y=x^2-1$ .
Find the coordinates of the points where $C_1$ and $C_2$ intersect.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).