1 Whole Numbers

1.1 Rounding Numbers

TrillionsPeriod

BillionsPeriod

MillionsPeriod

ThousandsPeriod

OnesPeriod

Hundre

dT

rillio

ns

Ten

Tri

llio

ns

Tri

llio

ns

Hundre

dB

illion

s

Ten

Billion

s

Billion

s

Hundre

dM

illion

s

Ten

Million

s

Million

s

Hundre

dT

hou

sands

Ten

Thou

sands

Thou

sands

Hundre

d

Ten

s

Ones

Ex. 1. What is in the hundred thousands place in62, 407, 981?

Ex. 2. What digit is in the ten millions place in417, 290, 006?

Ex. 3. Round 602, 549, 961 to the specific place.(a) ten thousands (b) tens (c) millions (d) thousands (e) hundreds

Practice 1.1.1. Pg 10 - 11 # 1 - 7 odd, 39 - 53 odd

1

1.5 Order of Operations

Note. Order of Operations Agreement

Perform operations in the following order:

1. Grouping symbols, which include parentheses ( ), brackets [ ], braces { }, and fraction bars −

2. Exponents

3. Multiplication or division in order as these operations occur from left to right

4. Addition or subtraction in order as these operations occur from left to right

Ex. 4. 20− 3 · 6÷ 2 Ex. 5. 12 + 14÷ 2 · 5 Ex. 6. 33 − 5 · 3 + 24÷ 4

Ex. 7. 34 − 5(8− 2)÷ 6 + (7− 3)2 Ex. 8. 72 − {5 + 3[18÷ (2 + 7)]}

Practice 1.5.1. Pg 55 # 1, 5, 7, 11, 13, 17, 21, 29 - 45 odd

2

1.6 More with Formulas

Definition. A polygon is a closed plane figure whose sides are line segments that intersect only at theendpoints.

The perimeter of a polygon is the sum of the lengths of all sides of the polygon.

Ex. 9. A rectangle has a length of 42 meters and awidth of 19 meters. Find the perimeter.

Ex. 10. The sides of a triangle are 3 in, 4 in, and 5in. Find the perimeter of the triangle.

Practice 1.6.1. Pg 68 # 1 - 5 odd, 35, 37

Definition. Composite shapes are shapes formed by putting together two or more fundamental shapessuch as rectangles or parallelograms. A composite shape also can be a shape within a shape.

Ex. 11. Find the area of the following figure:

3

Ex. 12. A family decides to build a rectangular fence with the following dimension 15 ft by 30 ft usingwood pieces that cost $ 8 per foot. What would be the total cost of the project?

Definition. Rectangles are special forms of a more general class of figures called parallelograms, whichhave two pairs of parallel sides.

How do we find the area of a parallelogram?

Note. Finding the area of a parallelogram is the same as finding the area of a .

Ex. 13. Find the area of the following parallelogram.

Practice 1.6.2. Pg 69 # 7 - 11, 42, 43, 46

4

Definition. Volume is a measure of the amount of space inside a three-dimensional object. We measurevolume using cubic units.

Note. The formula for volume of a cube is

V =

*Volume is in three dimensions.

Ex. 14. Find the volume of the following cube

Practice 1.6.3. Pg 71 # 13, 15, 17, 47, 48. 49

5

2 Integers, Z

2.1 Introduction to Integers, Z

Note. The symbol for integers is Z, since it stands for the German word for integer, Zahlen.

Ex. 1. Express each amount as a positive or negative integer.

(a) Ian has $3,250 in savings (b) A submarine is at a depth of 250 feet.

Ex. 2. Use the figure on the right to answer the following questions.

(a) What is the maximum normal Jan temper-ature in Fairbanks, Alaska? (b) What is the mini-mum normal Jan temperature in Denver, Colorado?

Practice 2.1.1. Pg 93 # 7 - 21 odd

6

Let’s study now the number line:

Ex. 3. Using the number line above, determine the appropriate symbol < or > for the following problems.

(a) 4 2 (b) -3 2 (c) -5 -9

Note. Some helpful Rules:

• Any positive number is greater than any negative number.

• 0 is greater than any negative number.

• When two negative numbers are compared, the negative number closer to 0 is the greater number.

Practice 2.1.2. Pg 94 # 29 - 45 odd

Definition. The absolute value is a given number’s distance from zero on a number line.

Ex. 4. Find the absolute value of each number.

(a) 14 (b) -27 (c) -1347

Practice 2.1.3. Pg 94 # 47 - 67 odd

7

2.2 Adding Z

2.3 Subtracting Z

2.4 Multiplying and Dividing Z; Exponents; Square Roots

Ex. 5. Perform the indicated operations.

(a) 19 + 74 (b) -42 -39

(c) -15 -18 (d) 20 - 18

(e) -14 + 24

Ex. 6. Greg’s January closing balance on his credit card was −$1, 320. In Feb, he made one payment of$450 and another payment of $700. His finance charges were $19. Greg made purchases at Tommy’s for$32 and Petromart for $27. What is his Feb closing balance?

8

Ex. 7. (a) Suppose we have $50 in a checking account and we write a check for $52. What is the balance?

(b) Suppose we have a balance of −$20 and write a check of $13. What is the balance?

Definition. An additive inverses are two numbers whose sum is zero.

a+ (−a) = 0 where a, for now, is any Z.

Ex. 8. (a) -5 + 5 = (b) 3 + (-3) =

Ex. 9. Perform the indicated operations:

(a) 34− 31 (b) 34− (−31) (c) −34− (−31) (d) −34 + (−31)

9

Ex. 10. Perform the indicated operation:

(−20)+(−11) =

(−4)+13 =

20+18 =

(−19)+(−5) =

(−12)−(−1) =

(−20)−(−5) =

18+22 =

25+(−6) =

22−9 =

27+(−3) =

(−4)−27 =

(−12)+12 =

(−7)−(−16) =

(−2)+17 =

(−12)+9 =

11+15 =

11−(−17) =

9+12 =

3−(−3) =

(−15)−(−10) =

(−10)+24 =

11+(−17) =

4−22 =

(−9)−16 =

2−10 =

13+(−18) =

17−10 =

(−20)+17 =

18+(−12) =

(−20)+17 =

24+27 =

1+27 =

9+(−20) =

9−19 =

20+(−19) =

24−20 =

(−7)+18 =

19+(−1) =

4+28 =

(−6)+(−19) =

(−3)−(−1) =

23+(−11) =

(−4)−(−1) =

6−13 =

7+19 =

1−13 =

(−17)+20 =

2+15 =

(−15)+(−1) =

25+7 =

7+22 =

(−8)−23 =

(−5)−24 =

(−3)−(−1) =

(−3)−12 =

17−(−18) =

12+(−12) =

(−8)−27 =

11+22 =

(−6)+(−10) =

(−7)−0 =

(−3)−18 =

(−9)+28 =

(−19)−8 =

5−14 =

24+8 =

(−7)−20 =

26+19 =

28−28 =

(−10)−23 =

17+5 =

(−16)−9 =

27−(−20) =

1−2 =

(−7)−(−18) =

23+23 =

13+1 =

(−2)−30 =

(−20)+25 =

(−16)−(−20) =

10−30 =

(−4)−1 =

(−11)−(−1) =

1+19 =

21−(−12) =

(−5)−(−6) =

1+7 =

15−(−9) =

(−20)−(−19) =

7−(−19) =

10

Definition. Net: Money remaining after subtracting costs from revenue (money made minus moneyspent)

Cost: Money spent on production, operation, labor, and debts.

Revenue: Income (money made).

Profit: A positive net (when revenue is greater than cost).

Loss: A negative net (when revenue is less than cost).

The definition of net suggest the following formula

Net = Revenue - Cost or N = R− C

Ex. 11. The revenue for a small business was $42, 382, and the total costs were $42, 295. What was thenet? Was it a profit or loss?

Ex. 12. The surface temperature on the moon at lunar noon can reach 120◦C. During the lunar night,the temperature can drop to −190◦C. How much of a change in temperature occurs from day to night onthe moon?

Practice 2.4.1. Pg 112 # 39 - 53 odd

11

Ex. 13. (a) −5 · (−8) (b) (−4)(−9) (c) −10(−13) (d) 2(−4)(−3)

Ex. 14. (a) 36÷ 4 (b) −40÷ 8 (c)−28

−7(d)

54

−9

Practice 2.4.2. Pg 121 # 1 - 25 odd, 43 - 57 odd, 95 - 99 odd

an where a, for now, is any Z and n, for now, are only positive Z (Z+).

Ex. 15. (a) 52 (b) (−5)2 (c) (−5)3

12

Ex. 16. (a) −53 (b) −52 (c) −24

Practice 2.4.3. Pg 122 # 27 - 41 odd

Definition. The square root of a given number is a number whose square is the given number.

Note. Every Z+ has two square roots, a positive square root and a negative square root with the sameabsolute value as the positive square root.

A negative number has no real square roots.

Ex. 17. Find all square roots of each number:

(a) 100 (b) 144 (c) −36 (d) −25

Here is a helpful table of perfect squares that could come in handy.

Note. The radical sign√

, for now, indicates the positive square root, or principal root.

Ex. 18. (a)√

121 (b)√

225 (c) −√

169 −√

361 (e)√−64

Practice 2.4.4. Pg 122 # 77- 93

13

2.5 Order of Operations

Note. Order of Operations Agreement

Perform operations in the following order:

1. Grouping symbols, which include parentheses ( ), brackets [ ], braces { }, absolute value | |,radicals

√, and fraction bars −

2. Exponents or roots in order as these operations occur from left to right

3. Multiplication or division in order as these operations occur from left to right

4. Addition or subtraction in order as these operations occur from left to right

Ex. 19. 33 − 4 · (−8) +√

81− |24÷ (−8)| Ex. 20. {−8 + 3[24÷ (4 + (−10))]}+ (−3)√

25 · 4

Practice 2.5.1. Pg 128 # 41 - 75 odd

14

2.6 Additional Applications and Problem Solving

Ex. 21. Maria bought 50 shares of stock at a price of $36 per share. One day day she sold 30 shares for$42 per share. The next day she sold the rest of her shares at a price of $30 per share. What was hernet? Was it a profit or a loss?

Ex. 22. Nicole put $1200 down when she bought her car. She made 48 payments of $350 and spent$2500 in maintenance and repairs. Four years after paying off the car, she sold it for $4400. What washer net? Was it a profit or loss?

Practice 2.6.1. Pg 134 #1 - 7 odd

15

3 Expressions and Polynomials

3.1 Translating and Evaluating Expressions

Definition. An expression is a constant, a variable; or any combination of constants, variables, andarithmetic symbols that describes a calculation.

An equation is a mathematical relationship that contains an equal sign.

Practice 3.1.1. Pg 150 # 1 - 11 odd

Note. Page 147 in your book has a table for Translating Basic Phrases. We will be using such phrases.

Ex. 1. Twelve more than eight times n. Ex. 2. The product of five and m subtracted fromnine.

Ex. 3. Sixteen less than the quotient of some number and two

Practice 3.1.2. Pg 151 # 13 - 29 odd

Ex. 4. Negative four times the difference of somenumber and nine

Ex. 5. The product of m and the square root of ndivided by the difference of h and k

Practice 3.1.3. Pg 151 # 31 - 35 odd

16

Evaluate the following:

Ex. 6. 5y − 7; y = −2 Ex. 7. 3p− 2q; p = 4, q = −3

Practice 3.1.4. Pg 151 # 39 - 59 odd

17

3.2 Introduction to Polynomials; Combining Like Terms

Definition. A monomial is an expression that is a constant or a product of a constant and variablesthat are raised to whole number powers.

Definition. A coefficient is the numerical factor in a monomial.

Definition. The degree of a monomial is the sum of the exponents of all variables in the monomial.

Definition. A polynomial is a monomial or a sum of monomials.

18

Definition. A polynomial in one variable is a polynomial in which every variable term has the samevariable. While a multivariable polynomial is a polynomial with more than one variable.

Definition. The degree of a polynomial is the greatest degree of any of the terms in the polynomial.

Definition. Like terms are constant terms or variable terms that have the same variables raised to thesame exponents.

Definition. Simplest form is an expression written with the fewest symbols.

Practice 3.2.1. Pg. 160 #5 - 27 odd

19

3.3 Adding and Subtracting Polynomials

Add and write the resulting polynomial in descending order for each of the following:

Ex. 8. (6t3 + 4t+ 2) + (2t4 + t+ 1) Ex. 9. (9m3+14m2+4m−8)+(2m3+3m2−4m−1)

Practice 3.3.1. Pg 166 #1 - 21 odd

Ex. 10. (6t3 + 4t+ 2)− (2t4 + t+ 1) Ex. 11. (9m3+14m2+4m−8)−(2m3+3m2−4m−1)

Practice 3.3.2. Pg 166 #27 - 45 odd

20

3.4 Exponent Rules; Multiplying Polynomials

Definition. am · an = am+n where m and n, for now, are positive Z.

Simplify the following:

Ex. 12. x3 · x4 Ex. 13. (n6)n Ex. 14. 2b3(−4b2)(3b5)

Practice 3.4.1. Pg 179 # 1 - 15 odd

Definition. (am)n = amn where m and n, for now, are positive Z.

Simplify the following:

Ex. 15. (3x2)4 Ex. 16. (−2m4)5 Ex. 17. (−2st3)6

Practice 3.4.2. Pg 179 # 21 - 33 odd

Simplify the following:

Ex. 18. 9(3x+ 1) Ex. 19. −5y(2y − 4) Ex. 20. −3a2(4a2 − a+ 3)

21

Ex. 21. (4a+ 1)(3a− 1) Ex. 22. (2m+ 7)(2m− 7)

Ex. 23. (a+ 2)(5a2 + 3a− 4) Ex. 24. −p3(p+ 2)(4p+ 1)

Practice 3.4.3. Pg 180 # 63 - 69 odd, 87 - 99 odd

22

Practice 3.4.4. Simplify the following problems:

(x+ 1)2 (4− 5t)2

Ex. 25. (m+ 5)2 Ex. 26. (4− n)2 Ex. 27. (3m+ 5n)2

23

3.5 Prime Numbers and GCF

Definition. A prime number is a natural number that has exactly two different factors; 1 and itself.

A composite number is a natural number that has factors other than 1 and itself.

Ex. 28. Determine whether the number is prime, composite, or neither:

39 119 1 107

Definition. Prime factorization is a product written with prime factors only.

Ex. 29. Find the prime factorization. Write your answer in exponential form.

64 560 910

Practice 3.5.1. Pg 191 #1 - 9 odd, 19 - 31 odd

24

Find the GCF using prime factorization for the following problems:

Ex. 30. 84 and 48 Ex. 31. 28 and 140

Ex. 32. 60 and 77 Ex. 33. 42, 63, and 105

Practice 3.5.2. Pg 192 # 41, 43, 51, 53 - 57 odd

Ex. 34. 12a5 and 15a Ex. 35. 48t5 and 60t8 Ex. 36. 20y3, 30y4, and 40y6

Practice 3.5.3. Pg 192 # 63 - 75 odd

25

3.6 Exponent Rules; Introduction to Factoring

Definition. am ÷ an =am

an= am−n where m and n, for now, are positive Z and a 6= 0.

Divide the following:

Ex. 37. 39 ÷ 32Ex. 38.

a12

a3Ex. 39.

t7

t7

Ex. 40. 10u7 ÷ (2u3) Ex. 41.28x3

−4x

Practice 3.6.1. Pg 201 #1 - 15 odd

26

Divide the following:

Ex. 42. (9x+ 21)÷ 3 Ex. 43. (18x4 − 12x3 + 6x2)÷ (−3x)

Note. To divide a polynomial by a monomial, divide each term in the polynomial by the monomial.

Practice 3.6.2. Pg 201 # 17 - 31 odd

Definition. Factored form is a number or an expression written as a product of factors.

Factor completely:

Ex. 44. 12y − 9 Ex. 45. 10x3 + 15x Ex. 46. 20a4 + 30a3 − 40a2

Practice 3.6.3. Pg 202 # 53 - 69 odd, 49, 51

27

3.7 Additional Application and Problem Solving

Ex. 47. Pg 208 #12

Definition. Surface area is the total number of square units that completely cover the outer shell of anobject.

28

Ex. 48. A carpenter builds wooden chests that are 2 feet wide by 3 feet long by 2 feet high. She finishedthe outside of the chests with stain. What is the surface area of the chest? What total area must sheplan to cover if she intends to put three coats of stain on the box?

Ex. 49. A business sells two types of small motors. The revenue from the sales of motor A and motor B isdescribed by 85A+105B+215. The total cost of producing the two motors is described by 45A+78B+345.Write an expression in simplest form for the net. In one month, the business sells 124 of motor A and119 of motor B. Using the expression for net, find the net profit or loss.

Practice 3.7.1. Pg 207 # 1, 9, 13, 17, 29, 31

29

4 Equations

4.1 Equations and Their Solutions

Definition. A solution is a number that makes an equation true when it replaces the variable in theequation.

Ex. 1. Determine if the given numbers is a solution for the given equation: 22 = 9b− 5.

0 1 3

Practice 4.1.1. Pg 227 #1 - 23 odd

30

4.2 The Addition Principle of Equality

Definition. A linear equation is an equation in which each variable term is a monomial of degree 1.

Note. The Addition Principle of Equality

Adding the same amount to both sides of an equation does not change the equation’s solution(s).

Solve the following:

Ex. 2. 16 = x− 5 Ex. 3. n+ 13 = 20 Ex. 4. 21 = −9 + k

Ex. 5. 4x+ 5− 3x = 12− 13 Ex. 6. 7h− 2(3h+ 4) = 15− 12

Practice 4.2.1. Pg 233 # 13 - 31 odd

31

Translate the following problems to an equation, then solve.

Ex. 7. A patient must receive 350 cubic centimeters of a medication in three injections. He has receivedtwo injections of 110 cubic centimeters each. How much medication should the patient receive in thethird injection?

Ex. 8. Thomas has a balance of −$568 on a credit card. How much must he pay to bring his balance to−$480?

Practice 4.2.2. Pg 234 # 33 - 43 odd

32

4.3 The Multiplication Principle of Equality

Note. The Multiplication Principle of Equality

Multiplying (or dividing) both sides of an equation by the same amount dose not change the equation’ssolution(s).

Solve the following:

Ex. 9. 5y = 45 Ex. 10. −2x = 16 Ex. 11. 9b+ 2 = 5b+ 26

Ex. 12. 3(3t− 5) + 13 = 4(t− 3)

Practice 4.3.1. Pg 243 # 7 - 29 odd

33

Translate the following problems to an equation, then solve.

Ex. 13. The length of a rectangle is 207 centimeters. Find the width of the rectangle if the perimeter is640 centimeters.

Ex. 14. A fish tank is to have a volume of 4320 cubic inches. The length is to be 20 inches, the width,12 inches. Find the height.

Practice 4.3.2. Pg 244 #35 - 45 odd

34

4.4 Translating Word Sentences to Equations

Note. Turn to page 247 for some of the phrases we use for Multiplication.

Ex. 15. Fifteen more than a number is forty. Ex. 16. Twelve times some number is negativeseventy-two.

Translate each problem and then solve.

Ex. 17. Eight less than the product of three and xis equal to thirteen.

Ex. 18. The difference of eight times b and nine isthe same as triple the sum of b and two.

Practice 4.4.1. 1 - 27 odd

35

4.5 Applications and Problem Solving

Definition. An isosceles triangle is a triangle with two sides of equal length.

An equilateral triangle is a triangle with all three sides of equal length.

Translate and solve:

Ex. 19. Suppose each of the equal length sides of an isosceles triangle is 3 inches more than the base.The perimeter is 30 inches. What are the lengths of the base and the sides of equal length?

Ex. 20. Two electric doors are designed to swing open simultaneously when activated. The design callsfor the doors to open so that the angle made between each door and the wall is the same. Because thedoors are slightly different in weight, the expression 15i− 1 describes the angle for one of the doors and14i + 5 describes the angle for the other door, where i represent the current supplied to the controllingmotors. What current must be supplied so that the door angles are the same?

36

Ex. 21. A marketing manager wants to research the sale of two different sizes of perfume. The smallbottle sells for $35; the large bottle, for $50. The report indicated that the number of large bottles soldwas 24 less than the number of small bottles sold and that the total sales were $4070. How many bottlesof each size were sold?

37

Ex. 22. A farmer has a total of 17 pigs and chickens. The combined number of legs is 58. How manypigs and how many chickens are there?

Practice 4.5.1. Pg 260 # 5, 7, 9, 11, 23 - 33 odd

38

5 Fractions and Rational Expressions

5.1 Introduction to Fractions

Definition. A fraction is a quotient of two numbers or expressions a and b having the forma

b, where

b 6= 0. Where a is call the numerator and b is call the denominator.

Definition. An improper fraction is a fraction in which the absolute value of the numerator is greaterthan or equal to the absolute value of the denominator.

Definition. A proper fraction is a fraction in which the absolute value of the numerator is less thanthe absolute value of the denominator.

Ex. 1. Margaret spends 8 hours each day at her office. Her company allows 1 of those hours to be alunch break. What fraction of her time at the office is spent at lunch? What fraction is spent working?

39

Definition. A mixed number is an integer combined with a fraction.

Ex. 2. Write each improper fraction as a mixed number:

13

3

32

5−37

4

−79

10

Ex. 3. Write the mixed numbers as an improper fraction.

105

6−11

2

919

Practice 5.1.1. Pg 282 # 1 - 9 odd, 17 - 31 odd, 41 - 47 odd

40

Ex. 4. Graph the following on a number line:2

3, −1

7, and

5

2.

Definition. An equivalent fraction is a fraction that name the same number.

Using the idea of equivalent fractions, determine the appropriate symbol <, >, or = for the followingproblems:

Ex. 5.3

7

4

9Ex. 6. 2

5

112

4

9Ex. 7. −2

7−12

42

Practice 5.1.2. Pg 284 # 55 - 59 odd, 69 - 79 odd

41

5.2 Simplifying Fractions and Rational Expressions

Definition. A fraction is in lowest terms when the greatest common factor of its numerator and de-nominator is 1.

Simplify to lowest terms

Ex. 8.30

40Ex. 9. −24

54Ex. 10.

28

6Ex. 11. −54

24

Ex. 12.9x4

24x3Ex. 13.

−15m4

20m2n

Practice 5.2.1. Pg 289 # 1 - 9 odd, 25, 31 - 35 odd, 39 - 49 odd

42

5.3 Multiplying Fractions, Mixed Numbers, and RationalExpressions

Multiply. Write the product in lowest terms.

Ex. 14.5

9· 3

10Ex. 15.

−18

20· 30

−32

Ex. 16. −41

8· 51

2Ex. 17.

(−2

3

5

)·(−7

1

4

)Ex. 18.

(−4

9

)· (−21)

Practice 5.3.1. Pg 303 # 9 - 31 odd

Multiply. Write the product in lowest terms.

Ex. 19.10a3

3· 12

aEx. 20.

18tu

5·(− 15t2

27u4

)

43

Practice 5.3.2. Pg 303 # 37 - 47 odd

Definition.(ab

)n=an

bnwhere n, for now, is a positive Z and b 6= 0.

Ex. 21.

(7x3

9

)2

Ex. 22.

(−m

3

2p

)5

Practice 5.3.3. Pg 303 # 49 - 59 odd

Ex. 23. A doctor says that 8 out of 10 patients receiving a particular treatment have no side effects. If ahospital gives 600 patients this treatment, how many can be expected to have no side effects? How manycan be expected to have side effects?

Ex. 24. A study estimates that2

3of all Americans own a car and that

1

4of these cars are blue. What

fraction of the population drives a blue car?

44

Ex. 25. Find the area of the triangle given:

Definition. A circle is a collection of points that are equally distant from a central point, called thecenter.

A radius is the distance from the center to any point on the circle.

A diameter is the distance across a circle along a straight line through the center. It can also be describeas twice the radius.

Definition. A circumference is the distance around a circle.

Definition. An irrational number is a number that cannot be expressed exactly as a fraction.

Definition. π is an irrational number that is the ratio of the circumference of a circle to its diameter.

45

5.4 Dividing Fraction , Mixed Numbers, and Rational Expressions

Definition. Reciprocals are two numbers whose product is 1.

Divide, Write the quotient in lowest terms.

Ex. 26.3

16÷ 5

12Ex. 27.

5

18÷(−2

9

)Ex. 28.

11417

Ex. 29. −1

5÷ (−4)

Ex. 30. 51

4÷ 2

1

3Ex. 31. −13

1

5÷(− 2

15

)

Ex. 32.8

13m÷ 12m

5Ex. 33. − 21c

30a3b÷(−14bc3

20a4

)

Practice 5.4.1. Pg 311 #13 - 33 odd

46

Simplify

Ex. 34.

√4

9Ex. 35.

√49

100

Practice 5.4.2. Pg 316 # 35 - 41 odd

Solve the following problems:

Ex. 36.3

4a =

2

3Ex. 37.

5

6y = − 3

10Ex. 38.

−4

15= −1

6n

Ex. 39. How many 41

2-ounce servings are in 30

1

4ounces of cereal?

Practice 5.4.3. Pg 316 # 43 - 59 odd

47

5.5 Least Common Multiple

Definition. The Least Common Multiple (LCM) is the smallest natural number that is divisible byall of the given numbers.

Find the LCM for each of the following:

Ex. 40. 18 and 24 Ex. 41. 21 and 35 Ex. 42. 12, 36, and 10

Ex. 43. 6a3 and 12a Ex. 44. 10hk5 and 8h2k

Practice 5.5.1. Pg 322 # 13 - 27 odd

48

Definition. The Least Common Denominator is the least common multiple of the denominator.

Write the equivalent fractions with the LCD

Ex. 45.9

20and

8

15Ex. 46.

2

3and

1

4

Ex. 47. −mn4

and3

mnEx. 48.

11

12t2and

7u

9tv

Practice 5.5.2. Pg 323 #33 - 43 odd

49

5.6 Adding and Subtracting Fractions, Mixed Numbers, andRational Expressions

Perform the indicated operations:

Ex. 49.−1

7+−3

7Ex. 50.

−7

16+

1

16Ex. 51. − 9

10−(− 1

10

)

Ex. 52.5t

u+

2

uEx. 53.

x+ 7

9+

2− x9

Note. To add or subtract fractions with different denominators:

• Write the fractions as equivalent fractions with a common denominator.

• Add or subtract the numerators and keep the common denominator.

• Simplify.

50

Perform the indicated operations:

Ex. 54.3

8+

1

6Ex. 55. − 7

18−(− 5

24

)

Ex. 56.5

6a+

3

4Ex. 57.

4

5y2− 1

5y

Ex. 58. 4 + 92

7Ex. 59. 5

5

6+ 7

1

3

Practice 5.6.1. Pg 334 # 23 - 41 odd

51

Subtract

Ex. 60. 79

10− 1

1

5Ex. 61. 10

1

5− 3

3

4

Ex. 62. 8− 51

4

Practice 5.6.2. Pg 335 #43 - 57 odd

52

Solve the following:

Ex. 63. a− 1

5=

5

8Ex. 64. m+ 3

1

2= −6

1

4

Ex. 65. A report claims that3

8of all cars are blue,

1

4are red, and the rest are other colors. What

fraction of all cars are other colors?

Practice 5.6.3. Pg 335 # 59 - 75 odd

53

5.7 Order of Operations; Evaluating and Simplifying Expressions

Note. Just because the following problems, from now on, have fractions does not mean that order ofoperations change. The rules still remain the same.

Ex. 66.

(2

5x3 − 1

3x2 − 5

6x+ 2

)−(

4x2 − 1

5x− 3 +

7

8

)

Definition. A trapezoid is a four-sided figure with one pair of parallel sides.

Note. The formula for the area of a trapezoid is A =1

2(a+ b)h.

Ex. 67. Find the area.

Note. The formula for the area of a circle with radius r is A = πr2.

Ex. 68. Find the area of a circle with a diameter of 7 inches.

54

Ex. 69. 71

4+

3

4

√4

9

Ex. 70. x+ vt; x = 64

5, v = 40, t =

1

6

Practice 5.7.1. Pg 345 # 1 - 13 odd, 19 - 47 odd

55

5.8 Solving Equations

Solve

Ex. 71. −7

9=

2

3n+

1

6Ex. 72.

1

4+

1

8m =

5

16

Ex. 73.3

5(x− 10) =

1

2(x+ 6)− 4

Practice 5.8.1. Pg 355 # 1 - 19 odd

56

Ex. 74. Translate to an equation; then solve:

3 more than3

4of k is equal to

1

8.

Ex. 75. Find the length a in the trapezoid shown if the area is 1221

2square inches.

Ex. 76. Two boards are joined. One board is2

3the length of the other. The two boards combine to be

71

2feet. What are the lengths of both boards?

Practice 5.8.2. Pg 356 #21 - 29 odd, 35 - 39 odd

57

6 Decimals

6.1 Introduction to Decimal Numbers

Definition. Decimal notation is a base-10 notation for expressing fractions.

Ex. 1. Write the word name:

0.91 − 0.602

124.90017

58

Ex. 2. Round 59.61538 to the specified place.

Tenths Hundredths Ten-thousandths

Practice 6.1.1. Pg 383 # 21, 23, 27, 33, 59 - 75 odd

Ex. 3. Write as a fraction or mixed number in simplest form:

0.79 − 0.08 2.6

Practice 6.1.2. Pg 383 # 1 - 19 odd

Ex. 4. Graph on a number line.

0.7 -6.21

-19.106

Ex. 5. Use < or > to write aa true statement.

0.61 0.65 45.192 45.092 − 0.04701 − 0.0471

Practice 6.1.3. Pg 384 # 39 - 57 odd

59

6.2 Adding and Subtracting Decimal Numbers

Add

Ex. 6. 14.103 + 7.035 Ex. 7. 0.1183 + 0.094 Ex. 8. 6.981 + 23

Subtract

Ex. 9. 146.79− 35.14 Ex. 10. 10.302− 9.8457

Ex. 11. 809− 162.648 Ex. 12. 1.987− 10.002

Practice 6.2.1. Pg 391 # 1, 5, 7, 11, 13, 17, 19, 21, 29, 33 - 43 odd

60

6.3 Multiplying Decimal Numbers; Exponents with Decimal Bases

Ex. 13. −1.5 · 2.91 Ex. 14. −6.45(−8) Ex. 15. (−0.6)3

Practice 6.3.1. Pg 402 # 1, 3, 9, 11, 17, 19, 37, 39, 43 - 51 odd, 59 - 63 odd

Definition. Scientific notation is a notation in the form a × 10n where a is a decimal number whoseabsolute value is greater than or equal to 1 but less than 10, and n is a Z.

Write each number in scientific notation.

Ex. 16. 91, 000 Ex. 17. −6, 390, 000 Ex. 18. −40, 913, 000, 000, 000

Practice 6.3.2. Pg 403 # 31 - 35 odd, 25 - 29 odd

61

6.4 Dividing Decimal Numbers; Square Roots with Decimals

Divide:

Ex. 19. 29÷ 5 Ex. 20. −13.49÷ 1.42

Ex. 21. 12.5÷ 3 Ex. 22. −605.3÷ 0.02

Ex. 23.3

4Ex. 24. −19

5

6

Practice 6.4.1. Pg 415 # 9 - 33 odd

62

Evaluate the square root

Ex. 25.√

0.01 Ex. 26.√

0.0016 Ex. 27.√

0.25

Practice 6.4.2. Pg 415 # 38, 40, 45, 49 - 63 odd

63

6.5 Order of Operations and Applications in Geometry

Note. The rules or Order of Operations for Decimals does not change, it still remains the same.

Ex. 28. [(6.2− 10)÷ 8]− 2√

1.21

Practice 6.5.1. Pg 428 # 1 - 13 odd

Definition. A Grade point average (GPA) is the sum of the total grade points earned divided by thetotal number of credit hours.

Below is the grade point to each letter grade.

A = 4 B+ = 3.5 B = 3 C+ = 2.5 C = 2 D+ = 1.5 D = 1 F = 0

Ex. 29. Calculate a student’s GPA rounded to the nearest thousandths.

Practice 6.5.2. Pg 428 #1 - 21 odd, 33 - 41 odd

64

Ex. 30. The volume of a cylinder is V = πr2h.

Ex. 31. The volume of a pyramid is V =1

3lwh.

Ex. 32. The volume of a cone is V =1

3πr2h.

Ex. 33. The volume of a sphere is V =4

3πr3.

65

Ex. 34. Find the volume of the afterburner fire cone from a jet engine.

Practice 6.5.3. Pg 433 # 53 - 67 odd

66

6.6 Solving Equations and Problem Solving

Ex. 35. 11− 2(5.6 + x) = 0.7x+ 1.3

Practice 6.6.1. Pg 444 # 1 - 23 odd

Definition. A right triangle is a triangle that has one right angle.

The hypotenuse is the side directly across from the 90◦ angle in a right triangle.

67

Ex. 36. In the construction of a roof, three boards are used to form a right triangle frame. An 8-footboard and a 6-foot board are brought together to form a 90◦ angle. How long must the third board be?

Ex. 37. Jack sold a mixture of 6-string electric and 12-string acoustic guitar strings to a customer. Heremembers that the customer bought 20 packs but doesn’t remember how many of each type. The pricefor a pack of 6-string electric strings is $6.95 and $10.95 for a pack of 12-string acoustic guitar strings. Ifthe total sale was $159, how many of each pack did the customer buy?

Practice 6.6.2. Pg 445 # 27, 29, 33, 39 - 43 odd

68

7 Ratios, Proportions, and Measurement

7.1 Ratios, Probability, and Rates

Definition. A ratio is a comparison between two quantities using a quotient.

For two quantities, a and b, the ratio of a to b isa

b.

Ex. 1. Stacie measure 40 inches from her feet to her navel and 24 inches from her navel to the top of herhead. Write the ratio of the smaller length to the larger length in simplest form. Write the ratio of thelarger length to the smaller length in simplest form.

Ex. 2. A recipe calls for 21

2cups of milk and

3

4of a cup of sugar. Write the ratio of milk to sugar in

simplest form.

Practice 7.1.1. Pg 470 # 1 - 9 odd

Definition. Theoretical probability of equally likely outcomes is the ratio of the number of favor-able outcomes to the total number of possible outcomes.

Ex. 3. If you have a regular six-sided die, a six-sided cube where each side contains a number from 1through 6. If you roll the die once, what is the probability of rolling a number that is greater than 4.

69

Ex. 4. What is the probability of selecting a 2 from a standard deck of cards?

Practice 7.1.2. Pg 471 # 13 - 23 odd

Definition. A unit ratio is a ratio in which the denominator is 1.

Ex. 5. A family has a total debt of $7842 and a gross income of $34, 240. What is its debt-to-incomeratio?

Definition. A rate is a ratio comparing two different measurements.

A unite rate is a rate in which the denominator is 1.

Ex. 6. Sasha drives 45 miles in3

4hour. After a break, she drives another 60 miles in

4

5hour. What was

her average rate for the entire trip in miles per hour?

70

Definition. A unit price is a unit rate relating price to quantity.

Ex. 7. A 50-ounce bottle of detergent costs $4.69. What is the unit price in cents per ounce?

Practice 7.1.3. Pg 472 # 27 - 45 odd

71

7.2 Proportions

Definition. A proportion is an equation in the forma

b=c

d, where b 6= 0 and d 6= 0.

2

7=

12

42

Note. If two ratios are proportional, their cross products are equal.

Practice 7.2.1. Pg 480 # 1 - 13 odd

Solve

Ex. 8.423

m=

34

−9Ex. 9.

−6.5

y=−13

14

Ex. 10. Erica drove 303.8 miles using 12.4 gallons of gasoline. At this rate, how much gasoline would ittake to drive 1000 miles?

72

Ex. 11. Xion can mow a 2500-square-foot lawn in 20 minutes. At this rate, how long will it take him tomow a 12,000-square-foot lawn?

Practice 7.2.2. Pg 481 # 15 - 37 odd

Definition. Similar figures are figures with the same shape, congruent corresponding angles, and pro-portional side lengths.

Ex. 12. Find the unknown lengths in the similar figures.

Practice 7.2.3. Pg 482 # 43 - 51 odd

73

7.3 American Measurement; Time

The following is a table of conversions we will be using in this course

Definition. A conversion factor is a ratio used to convert from one unit of measurement to another bymultiplication.

Ex. 13. 20 feet to inches Ex. 14. 43

10miles to feet

Ex. 15. 6000 square inches to square feet Ex. 16. 3 gallons to cups

Practice 7.3.1. Pg 491 # 5 - 13 odd, 21 - 53 odd

74

7.4 Metric Measurement

Suppose you want to go from meters to milimeters or from centiliters to liters. We use the followingprefixes to help us.

Ex. 17. 0.61 dekameter to decimeters Ex. 18. 9.74 meters to milimeters

Ex. 19. 456 meters to hectometers Ex. 20. 20,800 centimeters to kilometers

Definition. Mass is a measure of the amount of matter that makes up an object.

Ex. 21. 6500 miligrams to dekagrams Ex. 22. 0.15 kilograms to grams

Practice 7.4.1. Pg 500 # 3 - 23 odd

75

Ex. 23. 0.17 centiliter to cubic centimeters Ex. 24. 5910 g to metric ton

Ex. 25. 0.25 square meters to square centimeters Ex. 26. 95,100,000 square centimeters to squarehectometers

Practice 7.4.2. Pg 501 # 25 - 43 odd

76

7.5 Converting Between Systems; Temperature

American to Metric Metric to American

1 in = 2.54 cm 1 m ≈ 3.281 ft.1 yd. ≈ 0.914 m 1 m ≈ 1.094 yd.1 mi. ≈ 1.609 km 1 km ≈ 0.621 mi.1 qt. ≈ 0.946 L 1 L ≈ 1.057 qt.1 lb. ≈ 0.454 kg 1 kg ≈ 2.2 lb.1 T ≈ 0.907 t 1 t ≈ 1.1 T

First identify the direction of conversion, american to metric or metric to american, then use theappropriate conversion(s).

Ex. 27. 0.5 meter to inches Ex. 28. 412 feet to decimeter

Ex. 29. 40 ounces to grams

Practice 7.5.1. Pg 507 # 1 - 25 odd

Ex. 30. Suppose we want to know what 60◦C is in Fahrenheit.

77

Convert the following.

Ex. 31. −5◦C to degrees Fahrenheit

Ex. 32. −28◦F to degrees Celcius

Ex. 33. 40.1◦F to degrees Celcius

Practice 7.5.2. Pg 507 # 29 - 39 odd

78

7.6 Applications and Problem Solving

Ex. 34. A patient is to receive 250 cubic centimeters of 5% D/W solution IV over 4 hours. The label onthe IV bag indicates that 15 drops dissipate 1 cubic centimeter of the solution. How many drops shouldthe patient receive each minute?

Practice 7.6.1. Pg 515 # 17 - 21 odd

79

8 Percents

8.1 Introduction to Percent

Definition. A Percentage is a ratio representing some part out of 100.

Write each percent as a fraction in lowest terms.

Ex. 1. 48% Ex. 2. 6.25%

Practice 8.1.1. Pg 532 # 1 - 11 odd

Write each percent as a decimal number.

Ex. 3. 14% Ex. 4. 1.7% Ex. 5. 250%

Practice 8.1.2. Pg 532 # 13 - 23 odd

Write each fraction as a percent.

Ex. 6.3

5Ex. 7.

5

16Ex. 8.

4

9

Practice 8.1.3. Pg 532 # 25 - 35 odd

80

Write each decimal number as a percent.

Ex. 9. 0.49 Ex. 10. 0.883 Ex. 11. 0.13

Ex. 12. In a survey, 42% of the people rated a product as excellent, 34% rated it good, 14% rate it fair,and 10% rated it poor. Draw a circle graph that represents the results of the survey.

Practice 8.1.4. Pg 532 # 37 - 55 odd

81

8.2 Solving Basic Percent Sentences

A part is a percent of a whole.

Translate to an equation and solve.

Ex. 13. 30% of 148 is what number?

Ex. 14. What number is 120% of 48.5?

Practice 8.2.1. Pg 541 # 13 - 21 odd

82

Translate to an equation and solve.

Ex. 15. 45% of what number is 95.4?

Ex. 16. 801

4is 5% of what number?

Practice 8.2.2. Pg 542 # 25 - 33 odd

83

Translate to an equation and solve.

Ex. 17. What percent of 400 is 50?

Ex. 18. 15.37 is what percent of 14.5?

Practice 8.2.3. Pg 542 # 37 - 57 odd

84

8.3 Solving Percent Problems (Portions)

Translate to an equation; then solve.

Ex. 19. Sabrina score 80% on a test with 60 total questions. How many questions did she answercorrectly?

Ex. 20. Todd earned 20% commission on the sale of medical equipment. If Todd received $3, 857.79 asa commission, what was his total sale?

Ex. 21. Out of the 50 students taking foreign language at a certain school, 36 passed. What percentpassed?

Practice 8.3.1. Pg 547 # 1 - 31 odd

85

8.4 Solving Problems Involving Percent of Increase or Decrease

Solve the following.

Ex. 22. A suit is priced at $128.95. Find the sales tax and total amount if the tax rate is 6%.

Ex. 23. Benjamin has a current salary of $22, 600. If he receives a 3% raise, what is his new salary?

Ex. 24. A coat with an initial price of $88.95 is marked 40% off. Find the discount amount and finalprice.

Practice 8.4.1. Pg 560 # 3 - 13 odd

86

Solve the following.

Ex. 25. Julia indicates that she received a 5% raise in the amount of $2, 260. What was her formersalary?

Ex. 26. A college tuition from $4, 800 to $6, 000 per semester. What is the percent of increase in tuition?

Ex. 27. Josh recalls paying a total of $16.75 for a music CD He knows that the sales tax rate in his stateis 5%. What was the initial price of the CD?

Practice 8.4.2. Pg 561 # 17, 19

87

Solve the following.

Ex. 28. Marvin received a raise of $1, 728. If his former salary was $28, 800, what was the percent ofincrease?

Ex. 29. A salesperson says that she can take $1, 280 off the price of a car. If the initial price is $25, 600,what is the percent of decrease?

Practice 8.4.3. Pg 562 # 21 - 35 odd, 43 - 55 odd

88

8.5 Solving Problems Involving Interest

Definition. An interest rate is a percent used to calculate interest.

A principal is an initial amount of money invested or borrowed.

An interest is an amount of money that is a percent of the principal.

A Simple Interest is an interest calculated using only the interest rate, the original principal, and theamount of time in which the principal earns interest.

Note. Simple Interest Formula

I = Prt

I represents the simple interest

P represents the principal

r represents the simple interest rate

t represents the time in years

Find the simple interest and the final balance.

Ex. 30. $640 at 8% simple interest rate for 1 year. Ex. 31. $900 at 5.5% simple interest for 3 months.

Ex. 32. Ingrid notes that her savings account earned $42.32 in 1 year. If the account earns 6.2% simpleinterest, what was her principal?

Practice 8.5.1. Pg 573 # 1 - 13 odd

89

9 More with Geometry and Graphs

9.1 Points, Lines, and Angles

Definition. A point is a position in space having no length or width.

Definition. A line is a straight, one-dimensional figure extending forever in both directions. Ex :←→AB

Definition. A line segment is a straight, one-dimensional figure extending between two end points.Ex : AB

Definition. A ray is a straight, one-dimensional figure extending forever in one direction from a single

point. Ex :−→AB

Definition. A plane is a flat, two-dimensional surface extending forever in both dimensions.

Definition. Parallel Lines are lines lying in the same plane that do not intersect. Ex : l ‖ m

90

Definition. Perpendicular lines are two lines that intersect to form a 90◦ angle.

Note. For any triangle, the sum of the measures of the angels is 180◦.

Ex. 1. A triangle has the following angle measures (3x+ 5)◦, x◦, and (2x−5)◦. Find the measure of eachangle.

Practice 9.1.1. Pg 598 # 53 - 56

91

9.2 The Rectangular Coordinate System

Definition. The axis is a number line used to locate a point in a plane.

(a) (−4, 2) (b) (2,−4) (c)

(0,−3

1

2

)(d) (1, 0)

Ex. 2. Give the coordinates of each point.

Practice 9.2.1. Pg 608 # 1 - 19 odd

92

Note. Given two points with coordinates (x1, y1) and (x2, y2), the coordinate of their midpoint are(x1 + x2

2,y1 + y2

2

).

Ex. 3. Find the the midpoint of the given coordinate: (−2, 7) and (−5,−2).

Practice 9.2.2. Pg 609 # 21 - 33 odd

93

9.3 Graphing Linear Equations

Ex. 4. Is (-2, 13) a solution to the equation x+ y = 11?

Ex. 5. Is

(5, 2

1

3

)a solution to the equation y =

x

3+ 1?

Practice 9.3.1. Pg 618 # 1 - 11 odd

94

Ex. 6. Find three solutions for each equation:

(a) 3x+ y = 1 (b) 2x− 3y = 12 (c) y = 3

Ex. 7. Find the x- and y- intercepts for each on in the previous example.

Practice 9.3.2. Pg 618 # 41 - 59 odd

95

Ex. 8. Graph each line in Ex 6.

Practice 9.3.3. Pg 618 # 13 - 39 odd

96

9.4 Application with Graphing

Ex. 9. The linear equationp = 0.25r − 30, 000

describes the profit for a company, where r represent revenue.

(a) Find the profit if the revenue is $180, 000.

(b) Find the revenue required to break even.

Practice 9.4.1. Pg 630 # 3 - 7 odd

97

Ex. 10. The weekly cost to produce a toy is $0.75 per unit plus a flat $4, 500 for lease, equipment, supplies,and other expenses. Let C represent the total cost and u represent the number of units produced.

(a) Write a linear equation that describes the total cost.

(b) What would be the total cost to produce 600 units?

(c) What would be the total cost to produce 800 units?

Practice 9.4.2. Pg 630 # 9 - 10

98