Rational and modulus inequalities

What you’ll learn

• Solve rational inequalities with sign charts.
• Handle modulus expressions by intervals or justified squaring.

Understand the idea

Bring a rational inequality to one side and combine it into one fraction. Its sign can change at numerator zeros or denominator zeros, so these critical values divide the number line into intervals to test. Denominator zeros are always excluded.

A modulus measures distance and is non-negative. Split an expression at points where its inside changes sign, or square an inequality only after checking the signs of both sides. Multiplying by an expression of unknown sign without considering cases can reverse the required inequality.

Key formulae

$$|u|=\begin{cases}u&u\ge0\\-u&u<0\end{cases}$$

$$|x-a|<b\iff a-b<x<a+b\quad(b>0)$$

$$|x-a|>b\iff x<a-b\text{ or }x>a+b\quad(b>0)$$

Worked example

Solve $\frac{x-2}{x+1}\ge0$.

1. The critical values are $x=-1$ (undefined) and $x=2$ (zero).
2. Test the intervals: numerator and denominator have the same sign for $x<-1$ and $x>2$.
3. Include $2$ because equality is allowed; exclude $-1$ because the fraction is undefined.

Answer: $x<-1$ or $x\ge2$.