Basic Results for Inequalities

Basic Results

1. If \(a>b\) and \(b>c\), then \(a>c\). For example, \(12>5\) and \(5>-1\), so \(12>-1\).
2. If \(a<b\), then \(a\pm c<b\pm c\). For example, \(-5<7\), so \(-5\pm10<7\pm10\).
3. If \(a>b\) and \(c>0\), then \(ac>bc\) and \(\dfrac{a}{c}>\dfrac{b}{c}\). For example, \(21>9\), so \(\dfrac{21}{2}>\dfrac{9}{2}\).
4. If \(a>b\) and \(c<0\), then \(ac<bc\) and \(\dfrac{a}{c}<\dfrac{b}{c}\). For example, \(9<21\), so \(-\dfrac{9}{2}>-\dfrac{21}{2}\).
5. \(ab>0\) if and only if \(a<0\) and \(b<0\), or \(a>0\) and \(b>0\).
6. \(ab<0\) if and only if \(a<0<b\), or \(b<0<a\).
7. \(a<x<b\) if and only if \(a<x\) and \(x<b\).
8. “And” refers to intersection, while “or” refers to union.

Important Notes

Multiplying or dividing by a negative quantity reverses the inequality sign.