Quadratic Inequalities

Quadratics

Before we get started let’s review. A quadratic equation is an equation that canbe written in the form , where a, b and c are real numbers and a cannot

equalzero.

In this lesson we are going to discuss quadraticinequalities.

02 cbxax

Quadratic Inequalities

What do they look like? Here are some examples:

0732 xx

0443 2 xx

162 x

Quadratic Inequalities

When solving inequalities we are trying to find all possible values of the variablewhich will make the inequality true.

Consider the inequality

We are trying to find all the values of x for which the

quadratic is greater than zero or positive.

062 xx

Solving a quadratic inequality

We can find the values where the quadratic equals zero

by solving the equation, 062 xx

023 xx

02or03 xx

2or3 xx

1) (x+1)(x-2)>0

2) (x+2)(X-5)<0

4) X² < 5X - 6

5) X ² - 49 >0

6) 6x² < 24x

7) x² > 3x+8

Solving a quadratic inequality

You may recall the graph of a quadratic function is a parabola and the values we just found are the zeros or x-intercepts.

The graph of is

62 xxy

Solving a quadratic inequality

From the graph we can see that in the intervals around the zeros, the graph is either above the x-axis (positive) or below the x-axis (negative). So we can see from the graph the interval or intervals where the inequality is positive. But how can we find this out without graphing the quadratic?

We can simply test the intervals around the zeros in the quadratic inequality and determine which make the inequality true.

Solving a quadratic inequality

For the quadratic inequality,we found zeros 3 and –2 by solving the equation

. Put these values on a number line and we can see three intervals that we will test in the inequality. We will test one value from each interval.

062 xx

062 xx

-2 3

Solving a quadratic inequality

Interval Test Point

Evaluate in the inequality True/False

2,

3,2

,3

06639633 2

06600600 2

066416644 2

3x

0x

4x

True

True

False

062 xx

062 xx

062 xx

Solving a quadratic inequality

Thus the intervals make up the solution set for the quadratic inequality, .

In summary, one way to solve quadratic inequalities is to find the zeros and test a value from each of the intervals surrounding the zeros to determine which intervals make the inequality true.

062 xx ,3or2,

Example 2:

Solve First find the zeros by solving the equation,

0132 2 xx0132 2 xx

0132 2 xx

0112 xx

01or012 xx

1or2

1 xx

Example 2:

Now consider the intervals around the zeros and test a value from each interval in the inequality.

The intervals can be seen by putting the zeros on a number line.

1/2 1

Example 2:

Interval Test Point Evaluate in Inequality True/False

2

1,

1,2

1

,1

0x

4

3x

2x

0110010302 2

08

11

4

9

8

91

4

33

4

32

2

0316812322 2

False

True

False

0132 2 xx

0132 2 xx

0132 2 xx

Example 2:

Thus the interval makes up the solution set for

the inequality .0132 2 xx

1,2

1

Example 3:

Solve the inequality .

First find the zeros.

12 2 xx

012or12 22 xxxx

.formulaquadratictheusefactor,tdoesn'quadraticthisSince

22

12411 2

x4

71

4

71 i

Example 3:

But these zeros , are complex numbers.

What does this mean?

Let’s look at the graph of the quadratic,

4

71 ix

12 2 xxy

Example 3:

We can see from the graph of the quadratic that the curve never intersects the x-axis and the parabola is entirely below the x-axis. Thus the inequality is always true.

Example 3:

How would you get the answer without the graph?

The complex zeros tell us that there are no REAL zeros, so the parabola is entirely above or below the x-axis. At this point you can test any number in the inequality, If it is true, then the inequality is always true. If it is false, then the inequality is always false.

We can also determine whether the parabola opens up or down by the leading coefficient and this will tell us if the parabola is above or below the x-axis.

Summary

In general, when solving quadratic inequalities 1. Find the zeros by solving the equation you

get when you replace the inequality symbol with an equals.

2. Find the intervals around the zeros using a number line and test a value from each interval in the number line.

3. The solution is the interval or intervals which make the inequality true.

Practice Problems

02452 xx

012 2 xx

0116 2 x

0253 2 xx

0123 2 xx

06135 2 xx

09 2 x

0152 2 xx

452 xx

422 xx