Quadratic Inequalities

Quadratic inequalities are inequalities that involve a quadratic expression—an expression where the highest power of the variable is 2.

Example

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Solve for $x$ in the following inequality :

x^2-5x<-6

First, bring all the terms to one side :

x^2-5x+6<0

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

(x-2)(x-3)<0

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 :

$2>x>3$

$2<x<3$

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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^2-5x<-6

$x=1$

\begin{align*} (1)^2-5(1)&<-6 \\ 1-5 &< -6 \\ -4 & \not< -6 \end{align*}

Inequality is not satisfied for $x=1$ .

$x=2.5$

\begin{align*} (2.5)^2-5(2.5)&<-6 \\ 6.25-12.5 &< -6 \\ -6.25 & < -6 \end{align*}

Inequality is satisfied for $x=2.5$ .

$x=4$

\begin{align*} (4)^2-5(4)&<-6 \\ 16-20 &< -6 \\ -4 & \not< -6 \end{align*}

Inequality is not satisfied for $x=4$ .

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

2<x<3

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