Comparing functions with inequalities

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.