Quadratic Inequalities IES Sierra Nevada Algebra

Quadratic Inequalities

IES Sierra Nevada

Algebra

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.

062 xx

Solving a quadratic inequality

We can find the values where the quadratic equals zero

by solving the equation, 062 xx

Solving a quadratic inequality

Now, 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.

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, .It’s representation is:

062 xx ,32,

Summary

In summary, the steps for solving quadratic inequalities are:

1. Solve the equation.2. Plot the solutions on a number line creating

the intervals.3. Pick a number from each interval and test it in

the original inequality. If the result is true, that interval is a solution to the inequality.

4. Write properly the solution (the interval and the representation)

Example 2:

Solve First find the zeros by solving the equation,

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

0132 2 xx0132 2 xx

1or2

1 xx

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 .

Plot the solution!!

0132 2 xx

1,

2

1

Example 3:

Solve the inequality .

First find the solutions.

WHAT CAN WE DO NOW??

12 2 xx

012or12 22 xxxx

22

12411 2

x4

71

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