Solving Rational Inequalities Analytically

Exercise 5.

Find the solutions of the inequality

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


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


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

[Back] [Exercises]
[Algebra] [Trigonometry] [Complex Variables]
[Calculus] [Differential Equations] [Matrix Algebra]

S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Helmut Knaust

Copyright 1999-2019 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour