Quadratic Inequalities (Leaving Cert Mathematics): Revision Notes

Solve for $x$ in the following inequality :

First, bring all the terms to one side :

Next, factorise the quadratic equation using $(-b)$ formula, refer to quadratic factorisation.

So, our roots are $x=2$ and $x=3$

Now we need to test the intervals for which the $x$ values will satisfy the inequality, there are two possibilities :

To determine the correct interval, we do something called a region test. We pick three dummy values, one within the range of the roots, and two outside the range of the roots. Let's pick $1,2.5$ and $4$. Now we insert each of the values into the original inequality to see which range satisfies .

Original inequality :

$x=1$

Inequality is not satisfied for $x=1$.

$x=2.5$

Inequality is satisfied for $x=2.5$.

$x=4$

Inequality is not satisfied for $x=4$.

So the interval for which $x$ satisfies the inequality :