Solving Rational Inequalities Analytically

Find the solutions of the inequality

$\begin{displaymath}\frac{x+2}{x-1}\geq 1.\end{displaymath}$

Rewrite as

$\begin{displaymath}\frac{x+2}{x-1}-1\geq 0\end{displaymath}$

and simplify the left side:

$\begin{displaymath}\frac{x+2}{x-1}-1=\frac{(x+2)-(x-1)}{x-1}=\frac{3}{x-1}\geq 0.\end{displaymath}$

This inequality has only one critical point, namely x=1.

The set of solutions of the inequality is the set $(1,\infty)$ . x=1 is not a solution since it makes the denominator zero.

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Helmut Knaust
1998-06-16