To solve g(x) >= f(x), locate intersections and read where the graph of g is on or above that of f. Algebraically, solve g(x)-f(x) 0. Example x3 >= x becomes x(x-1)(x+1) 0. The critical values are -1,0,1; the signs from left to right are -,+,-,+. Thus -1 <= x 0 or x 1. For rational expressions, retain denominator exclusions and do not multiply by an expression with unknown sign. Use graphing technology for more complicated functions and check endpoints against the original inequality. A root of even multiplicity does not reverse the sign; an odd multiplicity does. A repeated root need not have multiplicity two.

To solve \(\mathrm{g}(x)\ge \mathrm{f}(x)\), locate intersections and read where the graph of \(\mathrm{g}\) is on or above that of \(\mathrm{f}\). Algebraically, solve \(\mathrm{g}(x)-\mathrm{f}(x)\ge0\).

\(x^3\ge x\) becomes \(x(x-1)(x+1)\ge0\). The critical values are \(-1,0,1\); the signs from left to right are \(-,+,-,+\). Thus \(-1\le x\le0\) or \(x\ge1\).

For rational expressions, retain denominator exclusions and do not multiply by an expression with unknown sign. Use graphing technology for more complicated functions and check endpoints against the original inequality.

A root of even multiplicity does not reverse the sign; an odd multiplicity does. A repeated root need not have multiplicity two.

Inequalities

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Comparing functions with inequalities

Comparing functions with inequalities