Solving Rational Inequalities Analytically

Exercise 5.

Find the solutions of the inequality

\begin{displaymath}\frac{x^2-9}{x+3}\leq 0.\end{displaymath}

Answer.

The numerator is zero for $x=\pm 3$, while the denominator is zero at x=-3; thus we have two critical points:

The tricky part is reading off all solutions. We have to exclude x=-3, since the expression on the left side of the inequality is undefined at x=-3, but points in the "neighborhood" of x=-3 are okay. So the set of solutions is the set $(-\infty,-3)\cup (-3,3]$.

Note that

\begin{displaymath}\frac{x^2-9}{x+3}=x-3,\end{displaymath}

only when $x\not=-3$. When x=-3, the left side is undefined, while the right side evaluates to -6.

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Helmut Knaust
1998-06-16

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