Modulus Inequalities

Core Modulus Results

For positive constants \(a\) and \(b\), with \(a<b\) in the compound case, the following results can be read directly from the graph of \(y=|x|\).

Examples

1. \(|x|\le10\) gives \(-10\le x\le10\).
2. \(|x|>10\) gives \(x>10\) or \(x<-10\).
3. \(|x|<-10\) has no solution.
4. \(|x|\ge-10\) gives \(x\in\mathbb{R}\).
5. \(-5<|x|<-10\) has no solution.
6. \(-5<|x|<2\) reduces to \(|x|<2\), so \(-2<x<2\).
7. \(2<|x|<7\) gives \(-7<x<-2\) or \(2<x<7\).

Important Notes

1. \(x^2=|x^2|=|x|^2\).
2. If both sides are non-negative, both sides can be squared: \(|a|<|b|\) if and only if \(a^2<b^2\).
3. Check the signs before squaring. For example, \(|x|<x-1\) has no solutions, whereas \(x^2<(x-1)^2\) gives \(x<\frac12\). Squaring without a sign check has introduced invalid values.