# Solve quadratic inequalities

Concept

To *solve quadratic inequalities*, we would normally use the graphical method to work out the solution. It would be advisable to first learn how to solve linear inequalities before coming back to this section.

For all quadratic inequalities of the form ,you first look at the coefficient a to see whether it is positive or negative. This determines the shape of the curve as seen below. Let's say for instance that we are asked to solve the quadratic inequality . The 'a' coefficient is positive , thus the shape of the curve is U shaped.

For purposes of clear explanation, you should add in a x axis to mark the points where your quadratic inequality meets the x axis. Then you need to solve .

By factorizing or using the equation, you will get the points x = -1 or x = 2.

Now , you need to look at whether the inequality sign is larger than ( >) or (<) .In this example, it's smaller than. So, we are looking at values of x where the curve is below the x-axis

Thus, the answer would be -1< x < 2 .

Examples

Solve the quadratic inequality

We go through the same systematic steps of solving quadratic inequalities.

1.Looking at the coefficient 3 (which is positive), we know that the curve is U shaped. We draw this out with a x axis across it.

2. Then we solve the equation and mark out the x values where the curve cuts the x axis.The values of x are-1/3 and 1.

3. We look at the values of x where the curve is equal or above the x axis. Thus , the correct answer would be or .

'Try it out yourself' Section

Try working out some quadratic inequalities below by yourself to get some practice.

1.

Spot the Trick Question

Sometimes, the examiner may try to trick the students (evil eh?) and twist the quadratic inequality question. There are times when the curve does not cut the x –axis, implying that its roots are imaginary.

Work it out by trying to solve for the value of x and you will obtain a negative value within the root sign. For instance,

In these situations, the correct answer would be no real solutions.

If you would like to share this article, feel free to syndicate it with a link to this article or to our algebra help site ,stating its ownership.Legal action will be taken against those who do not do so. For more information on linking, please go to 'link to us' link found below.