Intermediate Algebra Prerequisite Topics Review

• Quick review of basic algebra skills that you should have developed before taking this class

• 18 problems that are typical of things you should already know how to do

Order of Operations

• Many math problems involve more than one math operation

• Operations must be performed in the following order:– Parentheses (and other grouping symbols)– Exponents– Multiplication and Division (left to right)– Addition and Subtraction (left to right)

• It might help to memorize:– Please Excuse My Dear Aunt Sally

Example of Order of Operations

• Evaluate the following expression:

2

3

583

26431537

2

3

33

26121537

2

3

33

26337

93

86337

27

8697

27

8616

27

822

27

14

)separately simplified be should bottom and topsymbol; grouping a isbar fraction (A

Problem 1

• Perform the indicated operation:

• Answer:

212239

2130

7

30

Terminology of Algebra

• Expression – constants and/or variables combined in a meaningful way with one or more math operation symbols for addition, subtraction, multiplication, division, exponents and rootsExamples of expressions:

• Only the first of these expressions can be simplified, because we don’t know the numbers represented by the variables

32 x5n

104 wy 92

Terminology of Algebra

• If we know the number value of each variable in an expression, we can “evaluate” the expression

• Given the value of each variable in an expression, “evaluate the expression” means:– Replace each variable with empty parentheses– Put the given number inside the pair of parentheses

that has replaced the variable– Do the math problem and simplify the answer

Example

• Evaluate the expression for : 4,3 yx

2212

13

2212 xy

234212

9812

Problem 2

• Evaluate for x = -2, y = -4 and z = 3

• Answer:

z

yx

4

3 22

3

1

Like Terms

• Recall that a term is a _________ , a ________, or a _______ of a ________ and _________

• Like Terms: terms are called “like terms” if they have exactly the same variables with exactly the same exponents, but may have different coefficients

• Example of Like Terms:

constantvariable productconstant

variables

yxandyx 22 73

Simplifying Expressions by Combining Like Terms

• Any expression containing more than one term may contain like terms, if it does, all like terms can be combined into a single like term by adding or subtracting as indicated by the sign in front of each term

• Example: Simplify: xyxyx 26194yyxxx 21964

yx 219164 yx 179

head!your in done be

can steps twoMiddle

Simplifying an Expression

• Get rid of parentheses by multiplying or distributing

• Combine like terms

• Example:

xxxx 422253

xxxx 441053

143 x

Problem 3

• Simplify:

• Answer:

482327 mm

292 m

Linear Equations

• Linear equation – an equation where, after parentheses are gone, every term is either a constant, or of the form: cx where c is a constant and x is a variable with exponent1Linear equations never have a variable in a denominator or under a radical (square root sign)

• Examples of Linear Equations:

.

173 xx

xx5

332 1354 x

xxx 382

1627.

Solving Linear Equations

• Simplify each side separately– Get rid of parentheses– Multiply by LCD to get rid of fractions and decimals – Combine like terms

• Get the variable by itself on one side by adding or subtracting the same terms on both sides

• If the coefficient of the variable term is not 1, then divide both sides by the coefficient

Determine if the equation is linear. If it is, solve it:

427568 xx8273068 xx

12386 xx1223826 xxxx

1388 x38138388 x

378 x

8

37

8

8

x

8

37x

linear?it Is Yes

Problem 4

• Solve:

• Answer:

723295 xxx

19x

Linear Equations with No Solution or All Real Numbers as Solutions

• Many linear equations only have one number as a solution, but some have no solution and others have all numbers as solutions

• In trying to solve a linear equation, if the variable disappears (same variable & coefficient on both sides) and the constants that are left make a statement that is:– false, the equation has “no solution” (no number can

replace the variable to make a true statement)– true, the equation has “all real numbers” as solutions

(every real number can replace the variable to make a true statement)

Solve the Linear Equation

732 xxx

732 xxx

73 xx

73 xxxx

73 False!

solution no hasEquation

Solve the Linear Equation

xxxx 312272

True!

solutions are numbers real All

xxxx 62272

2828 xx

288288 xxxx

22

Problem 5

• Solve:

• Answer:

632

mmm

, Numbers, Real All -

Problem 6

• Solve:

• Answer:

xxxx 44623

Solution, No

Problem 7

• Solve:

• Answer:

6

2

3

5 xx

3

8x

Formulas

• A “formula” is an equation containing more than one variable

• Familiar Examples:LWA

2WL2P

Rectangle) a of (Area

Rectangle) a of (Perimeter

cba P triangle)a of (Perimeter

bh2

1A Triangle) a of (Area

Solving Formulas

• To solve a formula for a specific variable means that we need to isolate that variable so that it appears only on one side of the equal sign and all other variables are on the other side

• If the formula is “linear” for the variable for which we wish to solve, we pretend other variables are just numbers and solve as other linear equations

(Be sure to always perform the same operation on both sides of the equal sign)

Example

• Solve the formula for

ABA3

2

2

1

B:

ABA3

2

2

1

2

1

ABA

3

26

2

1

2

16

ABA 433

AABAA 34333

AB 3

33

3 AB

3

AB

Problem 8

• Solve for n:

• Answer:

niPA 1

Pi

PAn

Steps in SolvingApplication Problems

• Read the problem carefully trying to understand what the unknowns are (take notes, draw pictures, don’t try to write equation until all other steps below are done )

• Make word list that describes each unknown• Assign a variable name to the unknown you know the

least about (the most basic unknown)• Write expressions containing the variable for all the other

unknowns• Read the problem one last time to see what information

hasn’t been used, and write an equation about that• Solve the equation (make sure that your answer makes

sense, and specifically state the answer)

Example of Solving an Application Problem With Multiple Unknowns

• A mother’s age is 4 years more than twice her daughter’s age. The sum of their ages is 76. What is the mother’s age?

• List of unknowns– Mother’s age– Daughter’s age

• What else does the problem tell us that we haven’t used?Sum of their ages is 76

• What equation says this?

about?least know wedoWhich age sDaughter'x

42 x

7642 xx

Example Continued

• Solve the equation:

• Answer to question?

Mother’s age is 2x + 4:

7642 xx7643 x

476443 x723 x24x

4242 52

Example of Solving an Application Involving a Geometric Figure

• The length of a rectangle is 4 inches less than 3 times its width and the perimeter of the rectangle is 32 inches. What is the length of the rectangle?

• Draw a picture & make notes:

• What is the rectangle formula that applies for this problem?

width times3 than less inches 4 isLength

habout widt know Nothing inches 32 isPerimeter

WLP 22

Geometric Example Continued

• List of unknowns:– Length of rectangle:– Width of rectangle:

• What other information is given that hasn’t been used?

• Use perimeter formula with given perimeter and algebra names for unknowns:

width times3 than less inches 4 isLength

unknown basicmost theis This

43 xx

inches 32 isPerimeter

WLP 22

xx 243232

Geometric Example Continued

• Solve the equation:

• What is the answer to the problem?The length of the rectangle is:

xx 243232 xx 28632

8832 xx840

x5

45343x 11

Problem 9

• The perimeter of a certain rectangle is 16 times the width. The length is 12 cm more than the width. Find the width.

• Answer:.2 cmw

Inequalities

• An “inequality” is a comparison between expressions involving these symbols:

< “is less than”

“is less than or equal to”

> “is greater than”

“is greater than or equal to”

Inequalities Involving Variables

• Inequalities involving variables may be true or false depending on the number that replaces the variable

• Numbers that can replace a variable in an inequality to make a true statement are called “solutions” to the inequality

• Example:What numbers are solutions to:All numbers smaller than 5Solutions are often shown in graph form:

5x

0 5

)

thanlessmean tosparenthesi of use Notice

Graphing Solutions to Inequalities

• Graph solutions to:

• Graph solutions to:

• Graph solutions to:

• Graph solutions to:

0

0

0

0

2x

2x

2x

2x

2

2

2

2

]

)

[

(

Linear Inequalities

• A linear inequality looks like a linear equation except the = has been replaced by:

• Examples:

• Our goal is to learn to solve linear inequalities

or , , ,

137 xx

325

3 xx1354 x

xxx 382

1627.

Solving Linear Inequalities

• Linear inequalities are solved just like linear equations with the following exceptions:– Isolate the variable on the left side of the

inequality symbol– When multiplying or dividing by a negative,

reverse the sense of inequality– Graph the solution on a number line

Example of Solving Linear Inequality

04(

137 xx

13373 xxxx

172 x

71772 x

82 x

2

8

2

2

x

4x

Problem 10

• Solve and graph solution:

• Answer:

xxxx 741034

5x05

)

Three Part Linear Inequalities

• Consist of three algebraic expressions compared with two inequality symbols

• Both inequality symbols MUST have the same sense (point the same direction) AND must make a true statement when the middle expression is ignored

• Good Example:

• Not Legitimate:

.

142

13 x

142

13 x

142

13 x

Sense Same Havet Don' Symbols Inequality

1- NOT is 3-

Expressing Solutions to Three Part Inequalities

• “Standard notation” - variable appears alone in the middle part of the three expressions being compared with two inequality symbols:

• “Graphical notation” – same as with two part inequalities:

• “Interval notation” – same as with two part inequalities:

32 x

( ]2 3

]3,2(

SolvingThree Part Linear Inequalities

• Solved exactly like two part linear inequalities except that solution is achieved when variable is isolated in the middle

Example of SolvingThree Part Linear Inequalities

142

13 x

122

13 x

246 x

22 x SolutionNotation Standard

2 2

[ ) SolutionNotation Graphical

SolutionNotation Interval)2,2[

Problem 11

• Solve:

• Answer:

4123 m

2

32 m

22

3

[ ]

Exponential Expression

• An exponential expression is:

where is called the base and is called the exponent

• An exponent applies only to what it is immediately adjacent to (what it touches)

• Example:

nana

23x 3 not to x,only to appliesExponent 4m negative not to m, only to appliesExponent

32x (2x) toappliesExponent

Meaning of Exponent

• The meaning of an exponent depends on the type of number it is

• An exponent that is a natural number (1, 2, 3,…) tells how many times to multiply the base by itself

• Examples: 23x 4m

32x

xx3mmmm 1

xxx 222 38xexponentinteger any of meaning learn the willsection wenext In the

Rules of Exponents

• Product Rule: When two exponential expressions with the same base are multiplied, the result is an exponential expression with the same base having an exponent equal to the sum of the two exponents

• Examples:

nmnm aaa

24 33 243 63 47 xx 47x 11x

Rules of Exponents

• Power of a Power Rule: When an exponential expression is raised to a power, the result is an exponential expression with the same base having an exponent equal to the product of the two exponents

• Examples:

mnnm aa

243 243 83

47x 47x

28x

Rules of Exponents

• Power of a Product Rule: When a product is raised to a power, the result is the product of each factor raised to the power

• Examples:

nnn baab

23x 223 x 29x

42y 442 y 416y

Rules of Exponents

• Power of a Quotient Rule: When a quotient is raised to a power, the result is the quotient of the numerator to the power and the denominator to the power

• Example:

n

nn

b

a

b

a

23

x

2

23

x 2

9

x

Using Combinations of Rules to Simplify Expression with Exponents• Examples:

43225 pm 128425 pm 128165 pm 12880 pm

3325 yx 9635 yx 96125 yx

232332 32 yxyx 6496 98 yxyx 151072 yx

252

332

3

2

yx

yx

104

96

9

8

yx

yx

y

x

9

8 2

Integer Exponents

• Thus far we have discussed the meaning of an exponent when it is a natural (counting) number: 1, 2, 3, …

• An exponent of this type tells us how many times to multiply the base by itself

• Next we will learn the meaning of zero and negative integer exponents

• Examples: 0532

Definition of Integer Exponents

• The following definitions are used for zero and negative integer exponents:

• These definitions work for any base, , that is not zero:

10 an

n

aa

1

a

05 1 32

3

2

1

8

1

Quotient Rule for Exponential Expressions

• When exponential expressions with the same base are divided, the result is an exponential expression with the same base and an exponent equal to the numerator exponent minus the denominator exponent

Examples:

.

nmn

m

aa

a

7

4

5

5

4

12

x

x

374 55

8412 xx

“Slide Rule” for Exponential Expressions

• When both the numerator and denominator of a fraction are factored then any factor may slide from the top to bottom, or vice versa, by changing the sign on the exponentExample: Use rule to slide all factors to other part of the fraction:

• This rule applies to all types of exponents• Often used to make all exponents positive

sr

nm

dc

banm

sr

ba

dc

Simplify the Expression:(Show answer with positive exponents)

141

23

2

8

yy

yy

141

26

2

8

yy

yy

31

8

2

8

y

y

83

128

yy 11

16

y

Problem 12

• Evaluate:

• Answer:

42

16

1

Problem 13

• Evaluate:

• Answer:

10 33

3

4

Problem 14

• Use rules of exponents to simplify and use only positive exponents in answer:

• Answer:

22

2123

xy

yxx

62 yx

Polynomial

• Polynomial – a finite sum of terms

• Examples:

456 2 xx ?many terms How 3?first term of Degree

term?second oft Coefficien2

5-642 53 yxyx ?many terms How 2

term?second of Degree?first term oft Coefficien

103

Special Names for Certain Polynomials

Number of Terms

One term:

Two terms:

Three terms:

Special Name

monomial

binomial

trinomial456 2 xx

642 53 yxyx

yx29

Adding and Subtracting Polynomials

• To add or subtract polynomials horizontally:– Distribute to get rid of parentheses– Combine like terms

• Example:

233132 22 xxxxx

233132 22 xxxxx

xx 53 2

Multiplying Polynomials

• To multiply polynomials: – Get rid of parentheses by multiplying every

term of the first by every term of the second using the rules of exponents

– Combine like terms

• Examples:

4523 2 xxx 12156452 223 xxxxx 12112 23 xxx

4532 xx 1215810 2 xxx 12710 2 xx

Problem 15

• Multiply and simplify:

• Answer:

yxyx 234

22 328 yxyx

Squaring a Binomial

• To square a binomial means to multiply it by itself (the result is always a trinomial)

• Although a binomial can be squared by foiling it by itself, it is best to memorize a shortcut for squaring a binomial:

232x 3232 xx 9664 2 xxx 9124 2 xx

2ba 22 2 baba

232x

22 secondecond)2(first)(sfirst

9124 2 xx

Problem 16

• Multiply and simplify:

• Answer:

25 yx

22 1025 yxyx

Dividing a Polynomial by a Monomial

• Write problem so that each term of the polynomial is individually placed over the monomial in “fraction form”

• Simplify each fraction by dividing out common factors xyxyxyyx 224128 23

xyxy

xy

xy

xy

xy

yx

2

2

2

4

2

12

2

8 23

xyyx

1264 2

Problem 17

• Divide:

• Answer:

y

yyy

2

10468 23

yyy

5234 2

Dividing a Polynomial by a Polynomial

• First write each polynomial in descending powers

• If a term of some power is missing, write that term with a zero coefficient

• Complete the problem exactly like a long division problem in basic math

Example 415032 232 xxx

40150023 223 xxxxx

150023 23 xxxx3

xxx 1203 23 402 xx

( )

xx 122 2 150

2

8 0 2 2 xx( )

15812 x

4

158122

x

x

Problem 18

• Divide:

• Answer:

243 3 xxx

2

261163 2

xxx