Chapter 2 Section 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Chapter Chapter 22Section Section 88

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley


Solving Linear Inequalities

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44

33

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2.82.82.82.8Graph intervals on a number line.Use the addition property of inequality.Use the multiplication property of inequality.Solve linear inequalities by using both properties of inequality.Solve applied problems by using inequalities.Solve linear inequalities with three parts.

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66


Solving Linear InequalitiesInequalities are algebraic expressions related by

< “is less than,” ≤ “is less than or equal to,”

> “is greater than,” ≥ “is greater than or equal to.”

We solve an inequality by finding all real number solutions of it. For example, the solution set {x | x ≤ 2} includes all real numbers that are less than or equal to 2, not just the integers less than or equal to 2.

Slide 2.8 - 3


Objective 11

Slide 2.8 - 4

Graph intervals on a number line.


Graph intervals on a number line.A good way to show the solution set of an inequality is by

graphing.

We graph all the real numbers belonging to the set {x | x ≤ 2} by placing a square bracket at 2 on a number line and drawing an arrow extending from the bracket to the left (to represent the fact that all numbers less than 2 are also part of the graph).

Some texts use solid circles rather than square brackets to indicate the endpoint is included in a number line graph. (Open circles are also used to indicate noninclusion, rather than parentheses, when the endpoint is not included in the graph.)

Slide 2.8 - 5


Graph intervals on a number line. (cont’d)The set of numbers less than or equal to 2 is an example of an

interval on the number line. To write intervals, we use interval notation. For example, the interval of all numbers less than or equal to 2 is written (−∞, 2].

Slide 2.8 - 6

The negative infinity symbol −∞ does not indicate a number, but shows that the interval includes all real numbers less than 2.

As on the number line, the square bracket indicates that 2 is part of the solution.

A parentheses is always used next to the infinity symbol. The set of real numbers is written as (−∞, ∞).


Write each inequality in interval notation and graph the interval.

EXAMPLE 1Graphing Intervals Written in Interval Notation on a Number Line

Solution:

Solution:

Slide 2.8 - 7

3x

2 4x

[3, )

(2,4]


Objective 22

Slide 2.8 - 8

Use the addition property of inequality.


Consider the inequality 2 < 5. If 4 is added to each side, the result is

a true sentence.

2 54 4

6 9

Use the addition property of inequality.A linear inequality in one variable can be written in the form

where A, B, and C are real numbers, with A ≠ 0.

Slide 2.8 - 9

Ax B C

5 2x 3 5y 2 5 10.k Linear Inequalities

Examples of linear inequalities in one variable are

(All definitions and rules are also valid for >, ≥, and ≤.)


Now subtract 8 from each side:

The result again is a true sentence. These examples suggest the addition property of inequality.

2 58 8 6 3.

Use the addition property of inequality. (cont’d)

Slide 2.8 - 10

For any real numbers A, B, and C, the inequalities

and have exactly the same solutions.

That is, the same number may be added to each side of an inequality without changing the solutions.

A B A C B C

As with the addition property of equality, the same number may be subtracted from each side of an inequality.


EXAMPLE 2

Solve the inequality; then graph the solution set.

Solution:

Using the Addition Property of Inequality

1 8 7 2x x

Slide 2.8 - 11

71 8 7 2 7xx x x 11 12x x

( ,3)


Objective 33

Slide 2.8 - 12

Use the multiplication property of inequality.


Now multiply by each side of 3 < 7 by the negative number −5.

Use the multiplication property of inequality.

Multiply each side of the inequality 3 < 7 by the positive number 2.

To get a true statement when multiplying each side by −5, we must reverse the direction of the inequality symbol.

The addition property of inequality cannot be used to solve an

inequality such as 4x ≥ 28. This inequality requires the

multiplication property of inequality.

Slide 2.8 - 13

6 2 3 72

True

15 75 3 5

False

15 75 3 5

True


Use the multiplication property of inequality. (cont’d)

The multiplication property of inequality also permits division of each side of an inequality by the same nonzero number.

In summary, the multiplication property of inequality has two parts.

For any real numbers A, B, and C, with C ≠ 0,1. if C is positive, then the inequalities

and have exactly the same solutions;2. if C is negative, then the inequalities

and have exactly the same solutions.

Slide 2.8 - 14

That is, each of an inequality may be multiplied by the same positive number without changing the solutions. If the multiplier is negative, we must reverse the direction of the inequality symbol.

A B AC BC

A B AC BC


EXAMPLE 3

Solution:

Using the Multiplication Property of Inequality

Slide 2.8 - 15

12r

2 2

r

6r

( ,6)



Objective 44

Slide 2.8 - 16

Solve linear inequalities by using both properties of inequality.


To solve a linear inequality, use the following steps.

Step 1: Simplify each side separately. Use the distributive property to clear parentheses and combine like terms on each side as needed.

Solve linear inequalities by using both properties of inequality.

Slide 2.8 - 17

Step 2: Isolate the variable terms on one side. Use the addition property of inequality to get all terms with variables on one side of the inequality and all numbers on the other side.

Step 3: Isolate the variable. Use the multiplication property of inequality to change the inequality to the form x < k or x > k, where k is a number.

Remember: Reverse the direction of the inequality symbol only when multiplying or dividing each side of an inequality by a negative number..


EXAMPLE 4

Solution:

Solving a Linear Inequality

5 2 7 5x x x 74 72 7 5xx xx

Slide 2.8 - 18

23 5 22x 3 7

3 3

x

7

3x

7,

3



EXAMPLE 5

Solution:

Solving a Linear Inequality

4 1 3 15 2 1x x x

23 24 4x x xxx

Slide 2.8 - 19

3 1

3 3

2x

4x

4,

3 4 44x



Objective 55

Slide 2.8 - 20

Solve applied problems by using inequalities.


Inequalities can be used to solve applied problems involving phrases that suggest inequality. The table gives some of the more common such phrases, along with examples and translations.

Solve applied problems by using inequalities.

Slide 2.8 - 21

We use the same six problem-solving steps from Section 2.4, changing Step 3 to “Write an inequality” instead of “Write an equation.”

Do not confuse statements such as “5 is more than a number” with phrases like “5 more than a number.” The first of these is expressed as 5 > x, while the second is expressed as x + 5 or 5 + x.


EXAMPLE 6

Solution: Let x = Maggie’s fourth test score.

Finding an Average Test Score

98 86 8890

4

x

Slide 2.8 - 22

272 3 7260 2x

88x

272

44 4

x Maggie must get greater

than or equal to an 88.

Maggie has scores of 98, 86, and 88 on her first three tests in algebra. If she wants an average of at least 90 after her fourth test, what score must she make on that test?


Objective 66

Slide 2.8 - 23

Solve linear inequalities with three parts.


Inequalities that say the one number is between two other numbers are three-part inequalities. For example,

says that 5 is between −3 and 7.

Solve linear inequalities with three parts.

For some applications, it is necessary to work with a three-part inequality such as

where x +2 is between 3 and 8. To solve this inequality, we subtract 2 from each of the three parts of the inequality.

Slide 2.8 - 24

3

3 2x

3 22 2 2x

1 x


The idea is to get the inequality in the form

a number < x < another number,

using “is less than.” The solution set can then easily be graphed.

Solve linear inequalities with three parts. (cont’d)

Slide 2.8 - 25

When inequalities have three parts, the order of the parts is important. It would be wrong to write an inequality as 8 < x + 2 < 3, since this would imply 8 < 3, a false statement. In general, three-part inequalities are written so that the symbols point in the same direction and both point toward the lesser number.


Solve and graph the solution set.

EXAMPLE 7

Solution:

Solving Three-Part Inequalities

2 3 1 8x

2 3 11 1 8 1x

Slide 2.8 - 26

1,3

9

3 3

3 3

3

x

1 3x

Remember to work with all three parts of the inequality.

Chapter 2 Section 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

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Transcript of Chapter 2 Section 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.