2-5: Inequalities

© 2007 Roy L. Gover (www.mrgover.com)

Learning Goals:•Use interval notation•Solve linear and compound linear inequalities•Find exact solutions of quadratic and factorable inequalities

Important IdeaIn previous sections, we have been solving equalities, or equations. Now we are going to solve inequalities. The methods of solving equalities and inequalities are similar but there are important differences.

DefinitionThe statement c<d means that c is to the left of d on the number line.

d

c

DefinitionThe statement c>d means that c is to the right of d on the number line.

d

c

Important Idea

The statement c<d and d>c mean the same thing.

Definition

The statement b<c<d, called a compound inequality means:

b<c and simultaneously

c<d

Definition

,c d c x d

,c d c x d

Interval Notation:Let x,c & d be real numbers with c<d: ,c d c x d

,c d c x d

ExampleWrite the following using interval notation:2 5x 2 5x 3 8x

Try ThisWrite the following using interval notation:

3 8x

3,8

Try ThisWhat do you think this means?

19,

,0

Important IdeaPrinciples for solving inequalities:

1. Add or subtract the same number on both sides of the inequality.

Important IdeaPrinciples for solving inequalities:

2. Multiply or divide both sides of the inequality by the same positive number.

Important IdeaPrinciples for solving inequalities:

3. Multiply or divide both sides of the inequality by the same negative number and reverse the direction of the inequality.

Example

2 3 5 2 11x x

Solve. Write your answer using interval notation.

Try This

5 2 1 7x x

2,8

Solve. Write your answer using interval notation.

Example

4 3 5 18x

Solve. Write your answer using interval notation. Graph your answer on a number line.

Try ThisSolve. Write your answer using interval notation.Graph your answer on a number line.

2,23

2 4 3 6x

Important IdeaThe solutions of the form( ) ( )f x g x consist of

intervals on the x axis where the graph of f is below the graph of g.

Example( )f x

( )g x

( ) ( )f x g x

Important IdeaThe graph of ( ) ( )y f x g x lies above the x axis when ( ) ( )f x g x o and below the x axis when( ) ( )f x g x o

ExampleSolve:

4 3 210 21 40 80x x x x

Hint: Rewrite the inequality.

Try ThisSolve:

4 3 212 4 10x x x x

2.97x or4.21x

Important IdeaSolving an inequality depends only on knowing the zeros of the function and where the graph is above or below the x-axis. The zeros are where the function touches the x axis.

Example

Find the exact solutions:

2 6 0x x

Example

Find the exact solutions:22 3 4 0x x

Confirm with your calculator

Try This

2 3 2 0x x

Find the exact solutions:

3 17 3 17,

2 2

Example

Find the exact solutions:

Confirm with your calculator

6( 5)( 2) ( 8) 0x x x

Important IdeaSteps for solving inequalities:1. Write the inequality in one of these forms:

( ) 0f x ( ) 0f x

( ) 0f x ( ) 0f x

Important IdeaSteps for solving inequalities:2. Determine the zeros of f, exactly if possible, approximately otherwise.

Important IdeaSteps for solving inequalities:3. Determine the intervals on the x axis where the graph is above or below the x axis.

ExampleA store has determined the cost C of ordering and storing x laser printers.

300,0002C x

x

The delivery truck can bring at most 450 printers. How many should be ordered to keep the cost below $1600?

Lesson Close

Tell me everything you know about solving inequalities.