Nonlinear Inequalities
To solve nonlinear inequalities, we follow these steps:
- Transfer all terms to one side of the inequality sign and express the inequality in the form
or where is an expression in the variable (usually ) [to indicate its dependence on , we may write it as if we wish] - Factor
Q_{1},...,Q_{n} x P Q_{i} P \geq \leq ax+b ax^2+bx+c ax^2+bx+c a x^{2}+6>5x 5x x=2 x=3 (-\infty,2)\cup(3,\infty) x^{2} a=1 2 a x^{3}+2x^{2}\le x+2 x^{3}+2x^{2}-x-2 (x+2) x x=-2 x=-1 x=1 (x+2)(x-1)(x+1) x^{3}+2x^{2}-x-2 (-\infty,-2] [-1,1] x=-2 x=\pm1 \dfrac{x}{x+1}\leq2 x+1 x+1 (x+1) (-2-x)/(x+1) (-x-2) (x+1) x=-1 x=-2 x=-1 x=-1 (-x-2)/(x+1)\leq0 x=-2 x=-2 \dfrac{x+1}{x^{2}+1}>1 -1 (x^{2}+1) x^{2}+1>0 x(1-x) x=0 x=1 x(1-x)/(x^{2}+1)>0 0
