Inequality

An inequality compares two expressions using $<$ (less than), $\leq$ (at most), $>$ (greater than), or $\geq$ (at least). Unlike equations, inequalities typically have infinitely many solutions forming an interval or union of intervals.

Solving rules mostly mirror equations, with one critical exception: multiplying or dividing both sides by a negative number flips the inequality direction. For example, $-2x < 6$ becomes $x > -3$.

Compound inequalities like $-1 < 2x + 3 \leq 7$ are handled by performing operations on all three parts simultaneously. Quadratic inequalities ($x^2 - 4 > 0$) are solved by finding roots, then testing intervals between them.

Inequalities are essential for optimisation (linear programming), defining domains of functions, and bounding errors in numerical analysis.