Inequalities and Equations

[Pages:76]Inequalities and Equations

? 2011 Carnegie Learning

Tightrope walkers often perform at circuses. They have trained to keep their balance while walking across a thin, high rope. Some tightrope walkers use a large pole to help them

balance.

9.1 Call to Order Inequalities................................................................ 605

9.2 Opposites Attract to Maintain a Balance

Solving One-Step Equations Using

Addition and Subtraction............................................. 615

9.3 Statements of Equality Redux

Solving One-Step Equations Using

Multiplication and Division...........................................625

9.4 there are many ways . . . Representing Situations in Multiple Ways.....................635

9.5 Measuring Short

Using Multiple Representations

to Solve Problems...................................................... 643

9.6 Variables and More Variables The Many Uses of Variables in Mathematics.................653

9.7 Quantities that Change

Independent and Dependent Variables............................. 663

603

? 2011 Carnegie Learning

604 ? Chapter 9 Inequalities and Equations

Call to Order

Inequalities

Learning Goals

In this lesson, you will:

Use inequalities to order the number system. Graph inequalities on the number line.

Key Terms

inequality graph of an inequality solution set of an inequality ray

What happens every morning in your class and usually involves your teacher

calling names? If you said roll call, you'd be right! So, does your teacher seem to call your classmates' names in the same order every morning? Actually, there are a lot of ways for teachers to call roll, but one of the easiest ways is to call roll in alphabetical order. Sometimes teachers will call roll in alphabetical order in ascending order. This means starting at the letter A and moving to the letter Z. Or, teachers will call roll in alphabetical order in descending order, which is the opposite of ascending order.

Many people and items are ordered in different ways. When a photographer takes a picture of a group of people, the photographer will usually put the shorter people in the front of the group and the taller people in the back of the group. Mechanics usually arrange their wrenches and sockets in order from smallest to largest.

What things do you order? How do you go about ordering items or people--and this doesn't mean ordering your brother and sister around to do your chores!

? 2011 Carnegie Learning

9.1 Inequalities ? 605

Problem 1 Saying So Much with Just One Symbol

In the past, you probably used symbols that let you order numbers from least to greatest, or from greatest to least. These symbols are called inequality symbols. An inequality is any mathematical sentence that has an inequality symbol.

Symbol Meaning

Example

,

less than 3 , 5 3 is less than 5

.

greater than 10 . 7 10 is greater than 7

#

less than or equal to

3 # 9 3 is less than or equal to 9

$

greater than or equal to

4 $ 1 4 is greater than or equal to 1

fi

not equal to 6 fi 7 6 is not equal to 7

1. For each statement, write the corresponding inequality. a. 7 is less than or equal to 23 b. 56 is greater than 28 c. 2 is not equal to 5 d. 7.6 is less than 8.2 e. 5 _34_is greater than 4 _32_

? 2011 Carnegie Learning

606 ? Chapter 9 Inequalities and Equations

2. Write the meaning of each inequality in words. a. 7.8 fi 23.7

b. 8 _13_# 8.7 c._34_$ 0.75 d. 43,256 . 4489 e. 0.012 , 0.02

? 2011 Carnegie Learning

3. Write , or . to make each inequality true.

a. 12 2

b. 1.2 1.201

c. 3 _13_

3.3

d. 10.25

10 _15_

4. Write # or $ to make each inequality true.

a. 1 2

b. 4.2

c._13_

0.3

e. 24.33

24 _13_

d. 0.25

4 _14_ _25_

9.1 Inequalities ? 607

For any two numbers a and b, only one of the three statements is true.

a , b a . b a 5 b

5. What does this statement mean in terms of the ordering of the number system?

If a fi b, then a must be less than b or greater

than b.

Problem 2 Inequalities and the Number Line

A number line is a graphic representation of all numbers. 1. Plot and label each of the numbers shown on the number line. a. 3 b. 2.3 c. 3 _45_ d. 4 _13_ e. 4.66...

0

1

2

3

4

5

2. There are five points plotted on the number line shown. Identify the approximate location

of each point.

a

bc

d

e

0

1

2

3

4

5

a.

b.

c.

d.

e.

608 ? Chapter 9 Inequalities and Equations

? 2011 Carnegie Learning

3. A point at a is plotted on the number line shown. a

0 a. Plot a point to the right of this point and label it b. Then, write three different

inequalities that are true about a and b.

b. What can you say about all points to the right of point a on the number line?

4. A point at a is plotted on the number line shown. a

0 a. Plot a point to the left of this point and label it b. Then, write three different

inequalities that are true about a and b.

b. What can you say about all the points to the left of point a on the number line?

5. Describe the position of all the points on the number line that are:

a. greater than a.

b. less than a.

a

0

? 2011 Carnegie Learning

9.1 Inequalities ? 609

Problem 3 Graphing an Inequality on a Number Line

You can use a number line to represent inequalities. The graph of an inequality in one variable is the set of all points on a number line that make the inequality true. The set of all points that make an inequality true is the solution set of the inequality.

1. Look at the two inequalities x . 3 and x $ 3. a. Describe the solution sets for each.

b. Analyze the graphs of the two inequalities

shown on each number line.

Why does

x . 3 01234

one graph show a see-through point and the other one a

black point?

x $ 3

01234

Describe each number line representation.

c. How does the solution set of the inequality x $ 3 differ from the solution set of x . 3?

? 2011 Carnegie Learning

610 ? Chapter 9 Inequalities and Equations

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