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Section 1.7 Linear Inequalities and Absolute Value Inequalities

Section 1.7 Linear Inequalities and Absolute Value Inequalities

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Section 1.7 Linear Inequalities and Absolute Value Inequalities. Interval Notation. Example. Express the interval in set builder notation and graph:. Intersections and Unions of Intervals. Example. Find the set:. Example. Find the set:. Solving Linear Inequalities in One Variable. - PowerPoint PPT Presentation

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Page 1: Section 1.7 Linear Inequalities and Absolute Value Inequalities

Section 1.7Linear Inequalities

andAbsolute Value Inequalities

Page 2: Section 1.7 Linear Inequalities and Absolute Value Inequalities

Interval Notation

Page 3: Section 1.7 Linear Inequalities and Absolute Value Inequalities
Page 4: Section 1.7 Linear Inequalities and Absolute Value Inequalities
Page 5: Section 1.7 Linear Inequalities and Absolute Value Inequalities
Page 6: Section 1.7 Linear Inequalities and Absolute Value Inequalities

Example

Express the interval in set builder notation and graph:

3,2

0,4

,2

Page 7: Section 1.7 Linear Inequalities and Absolute Value Inequalities

Intersections and

Unions of Intervals

Page 8: Section 1.7 Linear Inequalities and Absolute Value Inequalities
Page 9: Section 1.7 Linear Inequalities and Absolute Value Inequalities
Page 10: Section 1.7 Linear Inequalities and Absolute Value Inequalities

Example

Find the set: 2,3 0,4

Page 11: Section 1.7 Linear Inequalities and Absolute Value Inequalities

Example

Find the set: 2,3 0,4

Page 12: Section 1.7 Linear Inequalities and Absolute Value Inequalities

Solving Linear Inequalities

in One Variable

Page 13: Section 1.7 Linear Inequalities and Absolute Value Inequalities

A linear inequality in x can be written in one of the following

forms: ax+b<0, ax+b 0, ax+b>0 ax+b 0. In each form

a 0.

Example:

-x+7 0

-x -7

x 7

When we multiply or divide b

oth sides of an inequality by

a negative number, the direction of the inequality symbol

is reversed.

Page 14: Section 1.7 Linear Inequalities and Absolute Value Inequalities
Page 15: Section 1.7 Linear Inequalities and Absolute Value Inequalities
Page 16: Section 1.7 Linear Inequalities and Absolute Value Inequalities

Example

Solve and graph the solution set on a number line:

4 5 7x x

Page 17: Section 1.7 Linear Inequalities and Absolute Value Inequalities

Checking the solution of a linear inequality on a Graphing Calculator

1

2

2 1

4

y x

y x

2 1 4x x

Y1=2x+1

Y2=-x+4

The region on the graph of the red box is where y1 is greater than y2. This is when x is greater than 1.

The intersection of the two lines is at (1,3). You can see this because both y values are the same, – 3.

The region in the red box is where the values of y1 is greater than y2.

Separate the inequality into two equations.

Page 18: Section 1.7 Linear Inequalities and Absolute Value Inequalities

Inequalities with

Unusual Solution Sets

Page 19: Section 1.7 Linear Inequalities and Absolute Value Inequalities

The solution set could be the null set, . The solution

set could be all real numbers, - , .

1

0 1

Never true

x x

1

0 1

Always true ,

x x

Page 20: Section 1.7 Linear Inequalities and Absolute Value Inequalities

Example

Solve each inequality:

3 4x x

Page 21: Section 1.7 Linear Inequalities and Absolute Value Inequalities

Solving

Compound Inequalities

Page 22: Section 1.7 Linear Inequalities and Absolute Value Inequalities

Now consider two inequalities such as

-3<2x+1 and 2x+1 3

express as a compound inequality

-3<2x+1 3

In this shorter form we can solve both inequalities

at once by performing the same operation on all t

hree

parts of the inequality. The goal is to isolate the x in

the middle.

Page 23: Section 1.7 Linear Inequalities and Absolute Value Inequalities

Example

Solve and graph the solution set on a number line.

3 1 2x

x

y

Page 24: Section 1.7 Linear Inequalities and Absolute Value Inequalities

Solving Inequalities

with Absolute Value

Page 25: Section 1.7 Linear Inequalities and Absolute Value Inequalities

The graph of the solution set for x >c will be divided

into two intervals whose union cannot be represented as

a single interval. The graph of the solution set for x

will be a single interval. Avoid

c

the common error of rewriting

x as -c<x>c.c

Page 26: Section 1.7 Linear Inequalities and Absolute Value Inequalities

Example

Solve and graph the solution set on a number line.

2 5 3x

Page 27: Section 1.7 Linear Inequalities and Absolute Value Inequalities

Example

Solve and graph the solution set on a number line.

2 5 3x

Page 28: Section 1.7 Linear Inequalities and Absolute Value Inequalities

Applications

Page 29: Section 1.7 Linear Inequalities and Absolute Value Inequalities

Example

A national car rental company charges a flat rate of $320 per week for the rental of a 4 passenger sedan. The same car can be rented from a local car rental company which charges $180 plus $ .20 per mile. How many miles must be driven in a week to make the rental cost for the national company a better deal than the local company?

Page 30: Section 1.7 Linear Inequalities and Absolute Value Inequalities

(a)

(b)

(c)

(d)

2 2 4x

Solve the absolute value inequality.

4 0

4 or x 0

4 and x 0

x 4 or x 0

x

x

x

Page 31: Section 1.7 Linear Inequalities and Absolute Value Inequalities

(a)

(b)

(c)

(d)

4 3 6 9x x Solve the linear inequality.

3

23

23

2

x

x

x

x