Safe way to clear a denominator containing x?

Multiply by the square of the denominator, which is always positive.

What are critical values?

The roots of the factors (and values where a denominator is zero) that split the number line into intervals.

Does (x+1)^2 change sign at x=-1?

No, a repeated root does not change the sign.

When can you square both sides of |f|g?

When both sides are non-negative, i.e. where g(x)\geq0.

What is an inequation?

An inequality containing an unknown, such as x^2-5x+60.

Algebraic inequalitiesEdexcel A-Level Further Maths: Flashcards

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What happens to an inequality when you multiply by a negative number?

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What happens to an inequality when you multiply by a negative number?: The inequality sign reverses.
Safe way to clear a denominator containing $x$ ?: Multiply by the square of the denominator, which is always positive.
What are critical values?: The roots of the factors (and values where a denominator is zero) that split the number line into intervals.
Solve $(x-2)(x+1)>0$ .: $x<-1$ or $x>2$ .
Does $(x+1)^2$ change sign at $x=-1$ ?: No, a repeated root does not change the sign.
$|f(x)|<k$ with $k>0$ means?: $-k<f(x)<k$ .
$|f(x)|>k$ with $k>0$ means?: $f(x)<-k$ or $f(x)>k$ .
When can you square both sides of $|f|>g$ ?: When both sides are non-negative, i.e. where $g(x)\geq0$ .
If $g(x)<0$ , what about $|f(x)|>g(x)$ ?: It is always true.
If $g(x)<0$ , what about $|f(x)|<g(x)$ ?: It is never true.
What is an inequation?: An inequality containing an unknown, such as $x^2-5x+6>0$ .
Solve $|x^2-1|>2(x+1)$ .: $x<-1$ or $x>3$ .

Exam questions on Algebraic inequalities

The inequality $\frac{x-1}{x+2}<2$ is to be solved, where $x\neq-2$ .
The student's method gives $x-1<2(x+2)$ , which leads to $x>-5$ . Show that $x=-3$ satisfies $x>-5$ but is not a solution of the original inequality, and explain the error.2 marks
The inequality $|2x-3|<x+4$ is to be solved.
Hence solve $|2x-3|\geq x+4$ .2 marks
Consider the inequality $\frac{3}{x-2}>\frac{2}{x+1}$ , where $x\neq2$ and $x\neq-1$ .
Show that the inequality is equivalent to $(x-2)(x+1)(x+7)>0$ .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).