Rational Inequalities (Leaving Cert Mathematics): Revision Notes

Solve for $x$ in the following inequality :

Following algebraic principles, it would be intuitive to multiply by $(x+2)$ to eliminate the fraction. However we don't know if the expression $(x+2)$ is positive or negative since it depends on what $x$ value we sub in. This means that we don't know whether or not to switch the inequality sign.

A simple way-around is to multiply both sides by $(x+2)^2$ because any expression squared is guaranteed to be positive, which means we don't have to switch the inequality sign :

We can rewrite the inequality as :

You should notice a common factor of $(x+2)$ which we can cancel out :

Expand and simplify :

Using the quadratic formula derives the roots : $x=3$ and $x=-2$. Evaluate the region test just as before :

So, the solution set is :