Why should you not multiply \frac{x}{x-3}2 by (x-3)?

x-3 may be negative, which would reverse the inequality; you would lose a case.

What can you multiply by safely in a rational inequality?

The square of the denominator(s), which is always positive.

Which values are critical values for a rational inequality?

Roots of the equation and values that make a denominator zero.

Are asymptote values ever in the solution?

No: the expression is undefined there.

When is squaring both sides of an inequality valid?

Only when both sides are non-negative.

Algebraic and modulus inequalitiesEdexcel International A Level Further Maths: Flashcards

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What is a critical value?

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What is a critical value?: A value of $x$ where the two sides are equal or an expression is undefined; the sign of the difference can only change there.
Solve $(x-2)(x-3)<0$ .: $2<x<3$
Solve $x^2>9$ .: $x<-3$ or $x>3$
What happens when you multiply an inequality by a negative number?: The inequality sign reverses.
Why should you not multiply $\frac{x}{x-3}>2$ by $(x-3)$ ?: $x-3$ may be negative, which would reverse the inequality; you would lose a case.
What can you multiply by safely in a rational inequality?: The square of the denominator(s), which is always positive.
Which values are critical values for a rational inequality?: Roots of the equation and values that make a denominator zero.
Are asymptote values ever in the solution?: No: the expression is undefined there.
$|f(x)|<g(x)$ is equivalent to what?: $-g(x)<f(x)<g(x)$
$|f(x)|>g(x)$ is equivalent to what?: $f(x)>g(x)$ or $f(x)<-g(x)$
Solve $|2x-3|<x+2$ .: $\frac13<x<5$
Solve $|x^2-1|>2(x+1)$ .: $x<-1$ or $x>3$
When is squaring both sides of an inequality valid?: Only when both sides are non-negative.

Exam questions on Algebraic and modulus inequalities

Consider the inequality $\frac{x}{x-3}>2$ .
Hence solve $\frac{x}{x-3}\le2$ .2 marks
Consider the inequality $|2x-3|<x+2$ .
Hence write down all the integers $x$ that satisfy $|2x-3|<x+2$ .2 marks
Consider the inequality $\frac{1}{x-1}>\frac{x}{x-4}$ .
Show that the inequality is equivalent to $(x-1)(x-4)\left(x^2-2x+4\right)<0$ .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).