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Equations and Inequalities in One Variable

- Chapter 2

Chapter Sections

2.1 Linear Equations The Addition and

Multiplication Properties of Equality 2.2

Linear Equations Using the Properties

Together 2.3 Solving Linear Equations Involving

Fractions and Decimals Classifying Equations 2.4

Evaluating Formulas and Solving Formulas for a

Variable 2.5 Introduction to Problem Solving

Direct Translation Problems 2.6 Problem

Solving Direct Translation Problems Involving

Percent 2.7 Problem Solving Geometry and

Uniform Motion Problems 2.8 Solving

Inequalities in One Variable

Linear Equations The Addition and

Multiplication Properties of Equality

- Section 2.1

Linear Equations in One Variable

A linear equation in one variable is an equation

that can be written in the form ax b c where

a, b, and c are real numbers and a ? 0.

3x 5 25

The expressions are called the sides of the

equation.

Solutions

The solution of a linear equation is the value or

values of the variable that make the equation a

true statement. The set of all solutions of an

equation is called the solution set. The

solution satisfies the equation.

Example Determine if x 1 is a solution to

the equation.

3(x 3) 4x 3 5x

3( 1) 3 4( 1) 3 5( 1)

3( 4) 4 3 5

12 12

True. x 1 is a solution

Solving Equations

Linear equations are solved by writing a series

of steps that result in the equation x a

number One method for solving equations is to

write a series of equivalent equations.

Two or more equations that have precisely the

same solutions are called equivalent equations.

3 5 8

1 7 2 6

Addition Property

The Addition Property of Equality states that for

real numbers a, b, and c, if a b, then a c

b c.

6 y 11

6 y (?6) 11 (?6)

y 5

Using the Addition Property

Example Solve the linear equation x ? 9 22.

Step 1 Isolate the variable x on the left side

of the equation.

x ? 9 9 22 9

Add 9 to both sides of the equation.

Step 2 Simplify the left and right sides of the

equation.

x 31

Apply the Additive Inverse Property.

Step 3 Check to verify the solution.

x ? 9 22

31 ? 9 22

22 22

?

Multiplication Property

The Multiplication Property of Equality states

that for real numbers a, b, and c, where c ?

0, if a b, then ac bc.

x 28

Using the Multiplication Property

Example Solve the linear equation 3x 81

Step 1 Get the coefficient of the variable x to

be 1.

Step 2 Simplify the left and right sides of the

equation.

x 27

Apply the Multiplicative Inverse Property.

Step 3 Check to verify the solution.

3x 81

3(27) 81

81 81

?

Linear Equations Using the Properties Together

- Section 2.2

Solving Linear Equations

Example Solve the linear equation 3x ? 9 ?24.

3x ? 9 ? 24

Add 9 to both sides of the equation.

3x ? 9 9 ? 24 9

3x ? 15

Divide both sides of the equation by 3

x ? 5

Check your answer in the original equation.

3(? 5) ? 9 ? 24

? 15 ? 9 ? 24

?

? 24 ? 24

Solving by Combining Like Terms

Example Solve the equation 3x ? 8 2x ? 15

3x ? 8 2x ? 15

3x 2x ? 7

Add 8 to both sides.

x ? 7

Subtract 2x from both sides.

3(? 7) ? 8 2(? 7) ? 15

Check your answer in the original equation.

? 21 ? 8 ? 14 ? 15

? 29 ? 29

?

Using the Distributive Property

Example Solve the equation 12 ? 2x ? 3(x 2)

4x 12 x.

12 ? 2x ? 3(x 2) 4x 12 x

12 ? 2x ? 3x ? 6 4x 12 x

Use the Distributive Property.

6 ? 5x 3x 12

Combine like terms.

6 8x 12

Add 5x to both sides.

6 8x

Subtract 12 from both sides.

Divide both sides by 8 and simplify.

Be sure to check your answer.

Summary Solving a Linear Equation

Summary Steps for Solving an Equation in

One Variable Step 1 Remove any parentheses

using the Distributive Property. Step 2 Combine

like terms on each side of the equation. Step 3

Use the Addition Property of Equality to get all

variables on one side and all constants on the

other side. Step 4 Use the Multiplication

Property of Equality to get the coefficient of

the variable to equal 1. Step 5 Check the

solution to verify that it satisfies the original

equation.

Solving Problems

Example A Chunky Cheeseburger contains 12 more

grams of fat than a Happy Hamburger. Find the

number of grams of fat in each sandwich if there

are a total of 98 grams of fat in the two

sandwiches by solving the equation x (x 12)

98 where x represents the number of fat grams

in a Happy Hamburger and x 12 represents the

number of fat grams in a Chunky Cheeseburger.

x (x 12) 98

2x 12 98

Simplify.

2x 86

Subtract 12 from both sides.

x 43

Divide both sides by 2.

There are 43 fat grams in a Happy Hamburger and

43 12 55 fat grams in a Chunky Cheeseburger.

Solving Linear Equations Involving Fractions and

Decimals Classifying Equations

- Section 2.3

Solving Equations Containing Fractions

Example Solve the equation

Fractions can be removed by multiplying both

sides of the equation by the LCD of all the

fractions.

The LCD is 10.

Remove all parentheses.

Divide out common factors.

Simplify.

Simplify.

Continued.

Solving Equations Containing Fractions

Example continued

Add 4 to both sides of the equation.

Divide both sides by 10

Check your answer in the original equation.

?

Solving Equations Containing Decimals

Example Solve the equation

Decimals can be eliminated by multiplying both

sides of the equation by the LCD of all the

decimals.

Distribute before eliminating the decimals.

Combine like terms.

Subtract 3.6x from both sides.

Subtract 5.4 from both sides.

Continued.

Solving Equations Containing Decimals

Example continued

Multiply both sides of the equation by the LCD 10

to eliminate the decimal.

Simplify.

Divide both sides by 38.

Check your answer in the original equation.

?

Conditional Equations Contradictions

A conditional equation is an equation that is

true for some values of the variable and false

for other values of the variable.

A contradiction is an equation that is false for

every replacement value of the variable.

Example Solve the equation 4x 8 4x 1.

4x 8 4x 1

8 1

Subtract 4x from both sides.

This is a false statement so the equation is a

contradiction. The solution set is the empty

set, written or ?.

Identities

An identity is an equation that is satisfied for

all values of the variable for which both sides

of the equation are defined.

Example Solve the equation 5x 3 2x 3(x

1).

5x 3 2x 3(x 1).

Distribute to remove parentheses.

5x 3 2x 3x 3

Combine like terms.

5x 3 5x 3

Subtract 5x from both sides.

3 3

This is a true statement for all real numbers x.

The solution set is all real numbers.

Evaluating Formulas and Solving Formulas for a

Variable

- Section 2.4

Simple Interest

A mathematical formula is an equation that

describes how two or more variables are related.

Interest is money paid for the use of money.

The total amount borrowed is called the principal.

The rate of interest, expressed as a percent, is

the amount charged for the use of the principal

for a given period of time, usually on a yearly

basis.

Simple Interest Formula If an amount of money, P,

called the principal is invested for a period of

t years at an annual interest rate r, expressed

as a decimal, the interest I earned is I

Prt This interest earned is called simple

interest.

Simple Interest

Example Jordan received a bonus check for 1000.

He invested it in a mutual fund that earned

6.75 simple interest. Find the amount of

interest Jordan will earn in two years.

The principal P is 1000.

The interest rate r is 6.75

The time t is 2.

Use the simple interest formula to solve the

problem

I Prt

(1000)(0.0675)(2)

135

Jordan will earn 135 interest at the end of two

years.

Geometric Formulas

Definitions The perimeter is the sum of the

lengths of all the sides of a figure. The area is

the amount of space enclosed by a two-dimensional

figure measured in square units. The surface area

of a solid is the sum of the areas of the

surfaces of a three-dimensional figure. The

volume is the amount of space occupied by a

figure measured in cubic units. The radius r of a

circle is the line segment that extends from the

center of the circle to any point on the

circle. The diameter of a circle is any line

segment that extends from one point on the circle

through the center to a second point on the

circle. The diameter is two times the length of

the radius, d 2r. In circles, we use the term

circumference to mean the perimeter.

Plane Figure Formulas

Continued.

Plane Figure Formulas

b

c

a

h

B

Solid Formulas

s

s

s

r

Continued.

Solid Formulas

Solving a Formula for a Variable

The formula for the area of a trapezoid is

Example Solve the formula for b.

Multiply both sides by 2.

Divide both sides by h.

Subtract a from both sides.

Evaluating a Formula

The formula for the area of a triangle is

Example If the area of a triangle is 66 inches,

and the base is 8 inches, find the height of the

triangle.

Substitute in the values.

Multiply both sides by 2.

Divide both sides by 8.

The height of the triangle is 16.5 inches.

Evaluating a Formula

Example Carla is making a planter out of an

empty can for her mothers birthday. She has 157

cubic inches of soil to use. Find the radius of

the can if it has a height of 8 inches.

The volume of a circular cylinder is V ?r2h.

Substitute the known values into the formula

V ?r2h

157 ?r2(8)

Simplify. (8? ? 25.13)

157 ? 25.13r2

6.25 ? r2

Divide both sides by 25.13.

Continued.

Evaluating a Formula

Example continued

6.25 ? r2

2.5 ? r

Take the square root of both sides.

The radius of the can is 2.5 inches.

Check Since r represents the radius of the

cylinder,

V ?r2h

?(2.5)2(8)

? 157

?

Introduction to Problem Solving Direct

Translation Problems

- Section 2.5

Translating English Expressions

Translating Sentences into Expressions

Example Translate each of the following into a

mathematical statement.

a.) Twelve more than a number is 25.

x 12

25

b.) One-third of the sum of a number and four

yields 6.

Problem Solving

Problem solving is the ability to use

information, tools, and our own skills to achieve

a goal.

The process of taking a verbal description of the

problem and developing it into an equation that

can be used to solve the problem is mathematical

modeling.

The equation that is developed is the

mathematical model.

Categories of Problems

- Five Categories of Problems
- Direct Translation problems that must be

translated from English into mathematics using

key words in the verbal description - Mixtures problems where two or more quantities

are combined in some fashion - Geometry problems where the unknown quantities

are related through geometrical formulas - Uniform Motion problems where an object travels

at a constant speed - Work problems problems where two or more

entities join forces to complete a job

Steps for Solving Problems

Solving Problems with Mathematical Models Step 1

Identify What You Are Looking For Read the

problem carefully. Identify the type of problem

and the information we wish to learn. Typically

the last sentence in the problem indicates what

it is we wish to solve for. Step 2 Give Names

to the Unknowns Assign variables to the unknown

quantities. Choose a variable that is

representative of the unknown quantity it

represents. For example, use t for time. Step 3

Translate into the Language of Mathematics

Determine if each sentence can be

translated into a mathematical statement. If

necessary, combine the statements into an

equation that can be solved.

Continued.

Steps for Solving Problems

Solving Problems with Mathematical Models Step 4

Solve the Equation(s) Found in Step 3 Solve

the equation for the variable and then answer the

question posed by the original problem. Step 5

Check the Reasonableness of Your Answer Check

your answer to be sure that it makes sense. If

it does not, go back and try again. Step 6

Answer the Question Write your

answer in a complete sentence.

Direct Translation Problem

Example In a baseball game, the Yankees scored 4

more runs than the White Sox. A total of 12 runs

were scored. How many runs were scored by each

team?

Step 1 Identify. This is a direct translation

problem. We are looking for the number of runs

scored by each team.

Step 2 Name. Let x represent the number of runs

scored by the White Sox. The number of runs

scored by the Yankees is equal to x 4.

Continued.

Direct Translation Problem

Example continued

Step 3 Translate. Since we know that the total

number of runs is 12, we have

Yankees runs

White Sox runs

x

x 4

12

Step 4 Solve.

x x 4 12

Combine like terms.

2x 4 12

2x 8

Subtract 4 from both sides.

x 4

Divide both sides by 2.

Continued.

Direct Translation Problem

Example continued

Step 5 Check. Since x represents the number of

runs scored by the White Sox, the White Sox

scored 4 runs. The Yankees scored x 4 4 4

8 runs.

4 8 12

?

Step 6 Answer the Question.

The Yankees scored 8 runs and the White Sox

scored 4 runs.

Problem Solving Direct Translation Problems

Involving Percent

- Section 2.6

Solving an Equation Involving Percent

Example A number is 9 of 65. Find the number.

Step 1 Identify We want to know the unknown

number.

Step 2 Name Let n represent the number.

Step 3 Translate

A number is 9 of 65

Continued.

Solving an Equation Involving Percent

Example continued

n 0.09 65

Step 4 Solve Solve the equation.

n 0.09 65

5.85

Step 5 Check Check the multiplication.

0.09 65 5.85

Step 6 Answer the Question

5.85 is 9 of 65.

Solving an Equation Involving Percent

Example 36 is 6 of what number?

Step 1 Identify We want to know a number.

Step 2 Name Let x represent the number.

Step 3 Translate

36 is 6 of what number

Continued.

Solving an Equation Involving Percent

Example continued

36 0.06x

Step 4 Solve Solve the equation.

36 0.06x

600

Divide both side by 0.06.

Step 5 Check Is 6 of 600 equal to 36?

0.06 600 36

36 36

?

Step 6 Answer the Question

36 is 6 of 600.

Solving a Business Problem

One type of percent problem involves discounts or

mark-ups that businesses use in determining their

prices.

Original Price Discount Sale Price Wholesale

Price Markup Selling Price

Example Julie bought a leather sofa that was

on sale for 35 off the original price. If she

paid 780, what was the original price of the

sofa?

Step 1 Identify This is a direct translation

problem. We are looking for the original price

of the sofa.

Step 2 Name Let p represent the original price.

Continued.

Solving a Business Problem

Example continued

Step 3 Translate The original price minus the

amount of the discount will equal the sale price.

p discount 780

The discount is 35 of the original price.

p 0.35p 780

Step 4 Solve Solve the equation.

Combine like terms.

0.65p 780

Divide both sides by 0.65.

p 1200

Continued.

Solving a Business Problem

Example continued

Step 5 Check If the original price of the couch

was 1200, then the discount would be 0.35(1200)

420. Subtracting 420 from the original price

of 1200 results in a sale price of 780.

Step 6 Answer the Question

The original price of the couch was 1200.

Problem Solving Geometry and Uniform Motion

- Section 2.7

Angles

Two angles whose sum is 90 are called

complementary angles. Each angle is called the

complement of the other.

Two angles whose sum is 180 are called

supplementary angles. Each angle is called the

supplement of the other.

Solving Angle Problems

- Example
- Angle A and angle B are complementary angles, and

angle A is 21º more than twice angle B. Find the

measure of both angles.

Step 1 Identify This is complementary problem.

We are looking for the measure of the two angles

whose sum is 90.

Step 2 Name Let a represent the measure of

angle A.

Continued.

Solving Angle Problems

- Example continued

Step 3 Translate Angle A is 21 more than twice

the measure of angle B.

a 21 2 m ? B

a 21 2 (90 a)

Step 4 Solve Solve the equation.

a 21 2 (90 a)

Distribute.

a 21 180 2a

Combine like terms.

a 201 2a

Continued.

Solving Angle Problems

- Example continued

a 201 2a

Add 2a to both sides.

3a 201

Divide both sides by 3.

a 67

Step 5 Check The measure of ? A is 67. The

measure of ? B is 90 a 90 67 23.

67 23 90

?

Step 6 Answer the Question

The two complementary angles measure 67 and 23.

Solving Triangle Problems

- Example
- Find the measure of Angle C.

Remember that the sum of the measures of the

interior angles of a triangle is 180.

Step 1 Identify This is an angles of the

triangle problem.

Step 2 Name Let c represent the measure of

angle C.

Continued.

Solving Triangle Problems

- Example continued

Step 3 Translate The three angles add up to

180.

84 42 c 180

Step 4 Solve Solve the equation.

84 42 c 180

Combine like terms.

126 c 180

Subtract 126 from both sides.

c 54

Continued.

Solving Triangle Problems

- Example continued

Step 5 Check Is the sum of the three angles

equal to 180?

84 42 54 180

180 180

?

Step 6 Answer the Question The measure of Angle

C is 54.

Solving Geometry Problems

- Example
- Julie is making cone-shaped candles. The mold

for the candles is 4 in. in diameter and 7 in.

high. How many cubic inches of wax does Julie

need to buy if she wants to make 50 candles?

Step 1 Identify This is a geometry volume

problem. We want to find the volume of the

cone-shaped candle to determine the amount of wax

needed for 50 candles.

Step 2 Name Let V represent the volume of one

cone.

Continued.

Solving Geometry Problems

- Example continued

Step 3 Translate We need to use the formula for

the volume of a cone.

? ? 3.14 r 2 h 7

Step 4 Solve Solve the equation.

Continued.

Solving Geometry Problems

- Example continued

This is the amount needed for one candle.

This is the amount needed for 50 candles.

29.3 in.3 ? 50 ? 1465 in.3

Step 5 Check

29.3 in.3 ? 50 1465 in.3

?

Step 6 Answer the Question Julie needs to buy

approximately 1465 in.3 of wax to make 50 candles.

Uniform Motion

Objects that move at a constant velocity (speed)

are said to be in uniform motion.

Uniform Motion Formula If an object moves at an

average speed r, the distance d covered in time t

is given by the formula d rt.

The following table is helpful in solving motion

problems.

Uniform Motion Problem

Example Steve jogs at an average rate of 8

kilometers per hour. How long would it take him

to jog 14 kilometers?

Step 1 Identify This is a uniform motion

problem. We are looking for the length of time

it would take Steve to jog 14 kilometers.

Step 2 Name Let t represent the length of time

it would take Steve to jog 14 kilometers.

Continued.

Uniform Motion Problem

Example continued

Step 3 Translate Organize the information in a

table.

d rt

14 8t

Step 4 Solve

Divide both sides by 8.

Simplify.

Continued.

Uniform Motion Problem

Example continued

Step 5 Check t 1.75 represents the length of

time.

d rt

14 (8)(1.75)

14 14

?

Step 6 Answer the Question

It takes Mark 1.75 hours (or 1 hour and 45

minutes) to run 14 kilometers.

Uniform Motion Problem

Example Nina drove her car to Cleveland while

Paula drove her car to Columbus. Nina drove 360

kilometers while Paula drove 280 kilometers.

Nina drove 20 kilometers per hour faster than

Paula on her trip. What was the average speed in

kilometers per hour for each driver?

Step 1 Identify

Distance problems can be solved using the

formula distance rate time (d rt).

Step 2 Name

Let r the rate of Paulas car.

Let r 20 the rate of Ninas car.

The time, t, for each driver was the same.

Continued.

Uniform Motion Problem

Example continued

Step 3 Translate

Step 4 Solve

Since the time for each driver was the same, we

can set the times equal to each other.

Continued.

Uniform Motion Problem

Example continued

Multiply both sides by r(r 20).

Simplify.

Distribute.

Subtract 280r from each side.

Divide each side by 80.

Paulas rate was 70 kilometers per hour. Ninas

rate was r 20 90 kilometers per hour.

Continued.

Uniform Motion Problem

Example continued

Step 5 Check

?

Step 6 Answer the Question

Paulas rate was 70 kilometers per hour. Ninas

rate was 90 kilometers per hour.

Solving Linear Inequalities in One Variable

- Section 2.8

Linear Inequalities

A linear inequality in one variable is an

inequality that can be written in the form ax

b lt c or ax b ? c or ax b gt c or ax b

? c where a, b, and c are real numbers and a ? 0.

Inequalities that contain one inequality symbol

are called simple inequalities.

means the set of all real numbers x greater

than four

x gt 4

Set-Builder Notation

Set-builder notation is used to express the

inequality in written form.

Representing an inequality on a number line is

called graphing the inequality, and the picture

is called the graph of the inequality.

x gt 4

Graphing Inequalities

Example Graph each interval.

Interval Notation

Interval notation is also used to represent

inequalities.

(4, ?)

x gt 4

Graph

Interval Notation

Set-Builder Notation

x x ? a

The interval a,??)

x x gt a

The interval (a, ??)

x x ? a

The interval (?, a

x x lt a

The interval (?, a)

x x is a real number

The interval (?,?)

Interval Notation

Example Write each inequality using interval

notation.

( 2, 0

( ?, 3)

( 1.5, 3)

Inequalities and Interval Notation

- SUMMARY Graphing Inequalities and Interval

Notation - If the inequality contains the symbol less than

or the symbol greater than. lt or gt, use

parentheses, ( or ), on the number line. - If the inequality contains the symbol less than

or equal to or the symbol greater than or equal

to, ? or ?, use brackets, or , on the

number line. - The symbols ? or ? always use parentheses.

Addition Property of Inequality

Addition Property of Inequality For real numbers

a, b, and c If a lt b, then a c lt b c

If a gt b, then a c gt b c

Example Solve the linear inequality and state

the solution set using set-builder notation and

interval notation. Graph the solution set.

x 24 gt 19

x gt 5

Subtract 24 from both sides.

Set-builder notation x x gt 5

Interval notation ( 5, ?)

Multiplication Property of Inequality

Multiplication Properties of Inequality Let a, b,

and c be real numbers. If a lt b, and if c gt 0,

then ac lt bc If a gt b, and if c gt 0, then

ac gt bc If a lt b, and if c lt 0, then ac gt bc

If a gt b, and if c lt 0, then ac lt bc

Example Solve the linear inequality and state

the solution set using set-builder notation and

interval notation.

2y lt 16

y gt ? 8

Divide both sides by ?2.

Set-builder notation y y gt 8

Interval notation ( 8, ?)

Solving Linear Inequalities

Steps for Solving Linear Inequalities Step 1

Remove parentheses. Step 2 Combine like terms

on each side of the inequality. Step 3 Get the

variable expressions on the left side of the

inequality and the constants on the right

side. Step 4 Get the coefficient of the

variable term to be one.

Solving Linear Inequalities

Example Solve the linear inequality and state

the solution set using set-builder notation and

interval notation. Graph the solution set.

Multiply by the LCD 24.

Simplify.

Distribute.

Simplify.

Continued.

Solving Linear Inequalities

Example continued

Subtract 8x from both sides.

Divide both sides by 5.

Set-builder notation x x ? 8

Interval notation ( ?, 8

Linear Inequality Problems

Linear Inequality Problems

Example Federation Express will not deliver a

package if its height plus girth (circumference

around the widest part) is more than 130 inches.

If you are preparing a package that is 33 inches

wide and 8 inches long, how high is the package

permitted to be?

Step 1 Identify. This is a geometry problem. We

are looking for the height of the package.

Step 2 Name. Let h represent the height.

Continued.

Linear Inequality Problems

Example continued

Step 3 Translate. The height plus the girth

cannot be more than 130 inches.

height circumference ? 130

h (2)(33) (2)(8) ? 130

Step 4 Solve.

h (2)(33) (2)(8) ? 130

Simplify.

h 66 16 ? 130

h 82 ? 130

Simplify.

Subtract 82 from both sides.

h ? 48

Continued.

Linear Inequality Problems

Example continued

Step 5 Check. Since h represents the height of

the package, check to make sure that the height

girth is not more than 130 inches.

h circumference ? 130

48 (2)(33) (2)(8) ? 130

48 66 16 ? 130

130 ? 130

?

Step 6 Answer the Question.

The height of the package cannot be more than 48

inches.

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