Introduction

Most, but not all, salaries and prices have soared over the decades. To make it easier to

compare, the figure below converts historical prices into today’s dollars, with adjustments based

on consumer’s price index.

The data for the average annual salary can be described by the mathematical model

262 12000S n

where S is the average U.S. annual salary, in dollars, n years after 1901.

If trends shown in the formula continue, when will the average annual salary be $50,000? Round

to the nearest year.

MATH 0030 – Lecture Notes

Section 2.1 The Addition Property of Equality

Section 2.2 The Multiplication Property of Equality

Solving an equation is the process of finding the ______________ or ___________________ that

make the equation statement _______________.

Circle the linear equations in one variable.

3 7 9x 23 7 9x 15

45x

15 45x 6.8x 6.8x

Example: Solve the equations.

a. 5 12x b. 2.8 5.09z c. 1 3

2 4x

Example: Solve by combining like terms before using the addition property.

8 7 7 10 6 4y y

Example: Solve by isolating the variable to one side of the equation.

3 6 2 5x x

Linear equation in one variable x is an equation that can be written in the form

ax b c

where a, b, and c are real numbers and a is not equal to 0

The Addition Property of Equality

The same real number (or algebraic expression) may be added to both sides of an equation

without changing the equation’s solution. This can be expressed symbolically as follows:

If a = b, the a + c = b + c

Example: Solve the equations.

a. 11 44y b. 15.5 5z c. 2

163

y d. 3x

Example: Solve using both the addition and multiplication properties.

a. 4 3 27x b. 4 15 25y c. 2 15 4 21x x

The Multiplication Property of Equality

The same nonzero real number (or algebraic expression) may multiply both sides of an

equation without changing the solution. This can be expressed symbolically as follows:

If a = b and 0c , then ac = bc.

Introduction:

In Massachussetts, speeding fines are determined by the formula

10( 65) 50F x

where F is the cost in dollars, of the fine if a person is caught driving x miles per hour.

Use this formula to solve, if a fine comes to $400, how fast was the person driving?

Example: Solve and check.

a. 7 25 3 16 2 3x x x b. 8 2( 6)x x c. 4(2 1) 29 3(2 5)x x

MATH 0030 - Lecture Notes

Section 2.3 Solving Linear Equations

Steps to Solving a Linear Equation.

1. Simplify the algebraic expression on each side.

2. Collect all the variable terms on one side and all the constant terms on the other side.

3. Isolate the variable and solve.

4. Check the proposed solution in the original equation.

Example: Solve by clearing fractions.

a. 3 39

2 5 5

x x b.

2 5

4 3 6

x x

0.3 10 0.37 100 0.408 1000

Example: Solve by clearing decimals.

a. 0.3 6 0.37 1.1x x b. 0.48 3 0.2 6x x

Example: Solve.

a. 3 7 3 1x x b. 3 1 9 8 6 5x x x

Clearing Equations with Fractions: We begin by multiplying _____________ sides of the

equation by the _________________________ of all fractions in the equation. The LCD is the

______________________ number that all the denominators will divide into. Multiplying

________________ term on both sides of the equation by the least common denominator will

eliminate the fractions in the equation.

Clearing Equations with Decimals: It is not necessary with a calculator, but multiplying a

decimal number by a power of _______________, has the effect of moving the decimal place to

the right.

Recognizing Inconsistent (No Solution) and Identities (All real Numbers)

If you attempt to solve an equation with no solution or one that is true for every real number,

you will eliminate the variable.

An inconsistent equation with NO SOLUTION results in a FALSE statement, such as 2 5

An identity that is true for ALL REAL NUMBERS results in a TRUE statement, such as 4 4

Solving for a Variable means rewriting the formula so that the variable is ____________________

on one side of the equation. It does not mean obtaining a numerical value for that variable.

Example: Solve the formulas for the indicated variables.

a. A lw solve for l b. 2 2P l w solve for l

c. PrS P t solve for r d. Ax By C solve for y

Example: Write an equation then solve.

a. What is 9% of 50?

b. 9 is 60% of what?

c. 18 is what percent of 50?

MATH 0030 - Lecture Notes

Section 2.4 Formulas and Percents

Formula’s involving Percent:

Percent Increase and Decrease

Percents are used for comparing changes, such as increases and decreases in sales, population,

prices, and production. If a quantity changes, its’ percent increase or percent decrease can be

determined

Example: Write an equation then solve.

a. A charity has raised $7500, with a goal of raising $500,000. What percent of the goal has

been raised?

b. Suppose that the state sales tax rate is 9% and you buy a graphing calculator for $96.

How much tax is due? What is the calculator’s total cost?

Percent Increase: The increase is what percent of the original amount?

Where A, represents the increase, P represents the percent and B represents the original

amount.

Percent Decrease: The decrease is what percent of the original amount?

Where A, represents the decrease, P represents the percent and B represents the original

amount.

Introduction

The bar graph show the ten most popular college majors with median, or middlemost, starting

salaries for recent college graduates.

The median starting salary for a nursing major exceeds that of a business major by $10 thousand.

Combined, their median starting salaries are $96 thousand. Determine the median starting

salaries of business and nursing majors with bachelor’s degrees.

Define the variables in words: x = ____________________________

Write and algebraic equation:

Solve:

Sentence Answer: _______________________________________________________________________

________________________________________________________________________________________

MATH 0030 - Lecture Notes

Section 2.5 An Introduction to Problem Solving

Strategy for Solving Word problems

1. Read the problem carefully. Define variables by letting x represent one of the unknown

quantities.

2. Write an equation in x that translates, or models the conditions of the problem.

3. Solve the equation and answer the problem’s question.

4. Check the solution in the original wording of the problem

English Phrase Algebraic

Expression

Addition

The sum of a number and 7

Five more than a number; a number plus 5

A number increased by 6: 6 added to a number

x + 7

x + 5

x + 6

Subtraction

A number minus 4

A number decreased by 5

A number subtracted from 8

The difference between a number and 6

The difference between 6 and a number

Seven less than a number

Seven minus a number

Nine fewer than a number

x – 4

x – 5

8 – x

x – 6

6 – x

x – 7

7 – x

x – 9

Multiplication

Five times a number

The product of 3 and a number

Two-thirds of a number (used with fractions)

Seventy-five percent of a number(used with decimals)

Thirteen multiplied by a number

A number multiplied by 13

Twice a number

5x

3x

2

3x

0.75x

13x

13x

2x

Division

A number divided by 3

The quotient of 7 and a number

The quotient of a number and 7

The reciprocal of a number

3

x

7

x

7

x

1

x

More than one operation

The sum of twice a number and 7

Twice the sum of a number and 7

Three times the sum of 1 and twice a number

Nine subtracted from 8 times a number

Twenty-five percent of the sum of 3 times a number and 14

Seven times a number, increased by 24

Seven times the sum of a number and 24

2x + 7

2(x+7)

3(1 + 2x)

8x – 9

0.25(3x + 14)

7x + 24

7(x+24)

Example: The bar graph shows that average rent and mortgage payments in the United States

have increased since 1975, even after taking inflation into account.

In 2008, rent payments averaged $824 per month. For the period

shown, monthly rent payments increased by approximately $7

per year. If this trend continues, how many years after 2008 will

rent payments average $929? In which year will this occur?

Define variables: x = __________________________

Equation:

Solve:

Sentence Answer: ________________________________________________________________________

__________________________________________________________________________________________

Example: A rectangular swimming pool is three times as long as it is wide. If the perimeter of

the pool is 320 feet, what are the pool’s dimensions?

Define variables: x = __________________________

Equation:

Solve:

Sentence Answer: ________________________________________________________________________

__________________________________________________________________________________________

Example: After a 40% price reduction, an exercise machine sold for $564. What was the

exercise machine’s price before this reduction?

Define variables: x = __________________________

Equation:

Solve:

Sentence Answer: ________________________________________________________________________

__________________________________________________________________________________________

MATH 0030 - Lecture Notes

Section 2.6 Problem Solving in Geometry

Geometric Formulas for Perimeter and Area

Square Area: 2A s where s is side

Square Perimeter: 4P s where s is side

Rectangle Area: A lw where l is length and w is width

Rectangle Perimeter: 2 2P l w where l is length and w is width

Triangle Area: 1

2A bh where b is base and h is height

Trapezoid Area: 1

2A h a b where b is base and h is height

Geometric Formulas for Circumference and Area of a Circle

Area of Circle: 2A r where r is radius

Circumference of Circle: 2C r where r is radius

Geometric Formulas for Volume

Cube Volume: 3V s where s is side

Rectangular Solid Volume: V lwh where l is length, w is width, and h is

height

Circular Cylinder Volume: 2V r h where r is radius and h is height

Sphere Volume: 34

3V r where r is radius

Cone Volume: 21

3V r h where r is radius and h is height

Circumference of Circle: 2C r where r is radius

Example: A sailboat has a triangular sail with an area of 24 square feet and a base that is 4 feet

long. Find the height of the sail.

Example: The diameter of a circular landing pad for helicopters is 40 feet. Find the area and

circumference of the landing pad. Express answers in terms of . Then round answers to the

nearest square foot and foot respectively.

Example: Which one of the following is the better buy: a large pizza with a 16-inch diameter for

$15.00 or a medium pizza with an 8-inch diameter for $7.50?

Example: A cylinder with a radius of 3 inches and a height of 5 inches has its height doubled.

How many times greater is the volume of the larger cylinder than the volume of the smaller

cylinder?

Example: In a triangle, the measure of the first angle is three times the measure of the second

angle. The measure of the third angle is 20 degrees less than the second angle. What is the

measure of each angle?

Example: The measure of an angle is twice the measure of its complement. What is the angle’s

measure?

Angles of a Triangle: The sum of the measures of the three angles of any triangle is

________________.

Complementary and Supplementary Angles

Complementary Angles are two angles having a sum of ____________________.

Supplementary Angles are two angles having a sum of _____________________.

Algebraic Expressions for Complements and Supplements

Measure of an angle: x

Measure of the angle’s complement: 90-x

Measure of the angle’s supplement: 180-x

Graphing solutions to linear inequalities are shown on a number line by shading all points

representing numbers that are solutions. SQUARE BRACKETS, [ ], indicate endpoints that

____________ solutions. PARENTHESES, ( ), indicate endpoints that ____________________

solutions.

English

Sentence

Inequality Interval

Notation

Set-Builder

Notation

Graph (number lines)

x is more than a x a

,a

x x a

x is at least a x a

,a x x a

x is less than a

x a

,a x x a

x is at most a x a

,a x x a

Example: Solve and graph the solution set on a number line.

a. 8 2 7 4x x b. 6 18x c. 5 3 17y

MATH 0030 - Lecture Notes

Section 2.7 Solving Linear Inequalities

Solving a Linear Inequality

1. Simplify the algebraic expression on each side.

2. Use the addition property of inequality to collect all the variable terms on one side and

all the constant terms on the other side.

3. Use the multiplication property of inequality to isolate the variable and solve. Change

the ___________________ of inequality when multiplying or dividing both sides by a

_________________ number.

4. Express the solution set in interval or set-builder notation, and graph the solution set on

a number line.

Example: Solve and graph the solution set on a number line.

a. 6 3 5 2x x b. 2 3 1 3 2 14x x

Example: Solve

a. 4 2 4 15x x b. 3 1 2 1x x x

Example: You can spend at most $1600 to have a party catered. The caterer charges $95 setup

fee and $35 per person. How many people can you invite while staying within your budget?

Recognizing Inequalities with No Solution or True for all Real Numbers

If you attempt to solve an inequality with no solution or one that is true for every real number,

you will eliminate the variable.

An inequality with no solution results in a _______________ statement, such as 0 >1. The

solution set is , the empty set.

An inequality that is true for every real number results in a ____________________

statement, such as 0<1. The solution set is , or x x is a real number .