NATURAL NUMBERS (counting numbers) 1, 2, 3, 4, 5,…web.cerritos.edu/imccance/SitePages/worksheets and syllabus/algebra... · NATURAL NUMBERS (counting numbers ... IRRATIONAL NUMBERS

1

NATURAL NUMBERS (counting numbers) 1, 2, 3, 4, 5,…

1 2 3 4 5 6 7 8 9 10

WHOLE NUMBERS 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …

0 1 2 3 4 5 6 7 8 9 10

INTEGERS…-3, -2, -1, 0, 1, 2, 3, …

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

RATIONAL NUMBERS Integers, Repeating and ending Decimals, and Fractions -3, -2 8

7, 0, 3, 5.7, 4.33333…

-5.5 0 2

1 1

IRRATIONAL NUMBERS

Decimals that don’t repeat or end. We don’t know exactly where they are on the number line. Like radicals, , e,

1.235698425624… there is no pattern.

REAL NUMBERS All of the previous numbers

RATIONAL INTEGERS WHOLE NUMBERS NATURAL

REAL

IRRATIONAL

So all natural numbers are whole numbers, all whole numbers are integers, all integers are rational, and all rational are real. The real

numbers are all the numbers on the real number line.

,3,0,12.234,5,.1,3,2

List all of the numbers that are:

1) whole numbers

2) Integers

3) Irrational

4) Rational

5) Real

2

Precisely graph the set of numbers on the following number line:

Absolute Value-

a) 3 b) 43

Opposite or -------- CHANGE SIGN

a) 34 b) 25 c) 5 d) 7 e) 32 f ) 555

Inequalities

< >

44 17<15 33

5___5 5_____5 6____6

Exponents

5x

52 42 42

02

3

8,

3

2,5.3,5.0,2,1

3

TESTS FOR DIVISIBILITY Multiples of 2

2 4 6 8 10 12 14 16 18 20 22

Multiples of 3

3 6 9 12 15 18 21 24 27 30 33

Multiples of 4

4 8 12 16 20 24 28 32 36 40 44

Multiples of 5

5 10 15 20 25 30 35 40 45 50 55

Multiples of 9

9 18 27 36 45 54 63 72 81 90 99

Multiples of 10

10 20 30 40 50 60 70 80 90 100 110 120

If a number can be divided by another number so that the remainder is 0, then we say:

1. The first number is DIVISIBLE by the second number. Ex. 6 is DIVISIBLE by 2

2. The second number DIVIDES the first. Ex. 2 DIVIDES 6

Tests for divisibility.

2 If a numbers last digit is a 0,2,4,6, or 8 then 2 divides that number.

Ex. 2 will divide 330 332 334 336 338 etc

3 If 3 divides the sum of a numbers digits then 3 divides that number.

Ex. The sum of 327’s digits is 3 + 2 + 7 = 12

3 will divide 12 , therefore 3 will divide 327.

5 If the number’s last digit is a 0 or 5 then 5 divides that number

Ex. 325 325’s last digit is a 5,

Therefore 5 will divide 325

2 Sum 3 5 7

234

721

100002

5210

125

4

))

))

bb

aa

6

4

8

3

4

6

8

3

32

1

24

5

32

1

24

5

____.__

____.__

____.__

3.012.0

0003.02.1

.

28.3313328.33.1

Different signs (one -) Subtract

Same signs (two – ) Add

Keep the sign of the larger

)5(454 = -1

)5(454 = -9

* 4343

145.362.0

1. Place the larger number on top. 1. Count the decimal places 1. Move the decimal for

2. Line up like place values and 2. Multiply. the divisor.

fill in missing place values. 3. Give the decimal places to 2. Place decimal on the line

the product. 3. Divide. Fill every place value

1. Find the LCD Convert all mixed numbers to improper 2

7

2

13

1. Reduce (cancel top to bottom) . 1.Flip the divisor (the second number)

2. Multiply and change the sign to multiply

LCD ________________________

2. Convert the denominators into the

LCD by multiplying each fraction

5

Order of operations

1. Parenthesis, Brackets, Absolute values. Perform all operations inside the parenthesis until you have one number.

2. Exponents, Radicals.

3. Multiplication or Division from LEFT TO RIGHT.

4. Addition or Subtraction from LEFT TO RIGHT.

1 2 3 4

_________________

_________________

_________________

_________________

_________________

Simplify completely

a) 9 2 1 4 11 b) 5 9 16

5 25

c) 236 2(11 14)

Simplify completely on the backside of the previous page.

a)20

7

15

22 b)

25 40 16

18 27 5

c)

1 38

6 4 d)

231

2 47

e) 9 2 1 4 11 f) 1 3

86 4

g) 236 2(11 14)

h) 5 9 16

5 25

i)

231

2 47 j) 25 40 16

18 27 5

k) 0 23 2 3 l) 217 7

m) )104()3(8 2 n) 80 o) 222 175

p) 2)9.0(37.4 q)

432

)2(410

r)

6

7

6

131

s) )21(432 t)

3)12(86 u) 524010 v) 0125

563

597

59428

53428

5234644

2

6

Undefined

060

6

edUndetermin

000

0

32

32

32

2x3x5=30

Changing a fraction to a Decimal. Round to the nearest thousandth.

7

32

Changing a fraction to a Percent. .

17

7

7

32

Changing a percent to a decimal.

%75.8

Changing a percent to a fraction.

%75.8

17

7

%35

%35

7

100

50

30

Two angles are COMPLEMENTARY ANGLES if_________________________________________

Draw an example:

mm

Two angles are SUPPLEMENTARY ANGLES if___________________________________________

Draw an example:

mm

VERTICAL ANGLES :__________________________________________________

Draw an example:

mm and mm

Two angles form a LINEAR PAIR if ____________________________________________________

Draw an example:

mm

The sum of a Linear pair of angles is____________________________________________

1. Find the missing angles.

8

135

a

a

b c

1–4. Find the missing measurements. a + b + c = 180

1. a = _____

2. b = _____

3. a = _____

135

a

105

4 a=

5. Find x and y.

a a

3 4 ° 6 6 °

a

1 4 ° b

y 60

60 x

9

h

2b

1b2

h

H

e

ig

ht

4 cm

7 cm

6 cm

4 cm

3 cm

3 cm

4 cm

4 cm

4 cm

4 cm

3 cm

Area= the number of unit squares in an object. (laying tile)

What is the area of this square?

Area = _____ Columns ______ Rows =________

Area = Base Height

Base

bhheightbaseA ramparallelog

21trapezoid2

1bbhA

Perimeter------

bhheightbaseA2

1

2

1triangle

10

Find the cost to tile the floor of each room with 12"X12" tiles, if the cost per tile is $4. The outline of your

rooms is below.

11

Evaluating

1) replace every variable with a ( )

2) Insert the given numbers for the given variables

abc 22 , a=3 b=-2 c=-5

Wrong Way Right Way

for a=-2, b=-1, c=3, and d=3

bb 32 , 15 abc acbdd 5

1

3

2

rtD 329

5 FC profit= revenue-cost Retail price = cost + markup

1) Find the distance covered by a jet if it travels for 3 hours at 550 mph.

2) Find the Celsius temperature reading if the Fahrenheit reading is 113 ?

3) For the month of June, a florist’s cost of doing business was $37,95. If June’s revenues totaled $5,115, what

was her profit for the month of June?

12

Properties

Name the property:

a) 3+5=5+3 b) 4(3+2) = 12+ 8 c) 3+0=3 d) 3x0=0

Expressions

Expressions do not have an equal sign. Expressions are like your hair sometimes there too long and you need to cut them down to size. You just simplify them.

Add or Subtract (Must have like terms)

+

1 + 3 4

The number in front tells you how many you have.

So, one apple plus three apples is four apples.

Remember this---the + means you can only count or remove your like terms, that is all. YOU CAN NOT CANCEL, OR USE EXPONENT RULES.

A) 5 + 3 B) 4 X + 7 X

C) 3 + 2 + 2 D) 8x - 5x² + 6x²

E) –m –n -8m +n G) 5

1 +

5

2 F) aAaAa 2812

13

1023 x

1) yxx 23 2) ABAB 39 3) 3X-8Y-10X+4

4) YY9

4

3

2 5)

22

12

5

8

3xx 6) xyx 92.325.0

Multiplying with variables Multiply the numbers and write the letters

x23 yx 32 34s yx 23 ysr 325

DISTRIBUTING

23 x 125 x 2 x 235 x 265 r

Expression #1 15423 xx It’s to long, cut it down to size.

Distribute

Combine like terms

Try:

)4(32243 xx xxx 783144

1

1023

46203

42063

x

xx

xx

14

Phrases

The following four phrases show operations between two things. The create quantities, parenthesis:

The difference of 4 and 3 The sum of 4 and 3 The product of 4 and 3 The quotient of 4 and 3

( 4 - 3) ( 4 + 3) ( 4 · 3) ( 4 3)

The product of 4 plus 3 and 7 minus a number. The product of 4 plus 3 and 7, minus a number

x 734 x 734

Example: four times the sum of five and six.

4 · ( 5 + 6 ) 4 (5 + 6)

TWO SPECIAL CASES

A number subtracted from 3 7 less than a number

X - 3 7 - X

3 - X X - 7

Translate each phrase.

1. The product of 3 and 4 2. The sum of 7 and 8 3. 4 subtracted from 8

4. The quotient of 6 and 3 5. 8 times the sum of 4 and 3 6. The sum of 4 times 2 and 5 times a #

7. 4 minus the sum of 5 and 2. 8. The difference of 6 and 4

9) The sum of 4 and the difference of 3 and a number10) The product of a number and the sum of 6 and the same number. 11) half of the quotient of 3 and a number 12) The quotient of x and the sum of x and 3.

13) The difference of 7 and the product of 4 and a number

14) The difference of the product of 3 and a number and the sum of 5 and the same number.

15

Warm-up:

23 X 42 X 153 X 32

X 42 X

Question: Is 1 a solution to 743 x ? ___________why?_____________________________________

Simple Equation

Isolate the x 1863 x (x’s on one side, non x’s on the other)

(add and subtract)

Divide (Divide by the number next to the variable)

Turning a Basic Equation into a Simple Equation

Isolate the x xx 1863 (pick a side to put the Xs on)

(add and subtract)

Divide (Divide by the number next to the variable)

16

SOLVING EQUATIONS-------Eliminate

S-I-D We want to know what x is for the equation to be true

We have to find x =

Simplify Isolate Divide

Simplify xx 1823 (simplify each side)

a) Remove fractions ----- Multiply both sides by the LCD

b) Remove parenthesis ----- Distribute. c) Remove decimals -----Multiply by powers of ten

d) Combine like terms

Isolate the x xx 1863 (x’s on one side, non x’s on the other)

(add and subtract)

Divide 122 x (Divide by the number next to the variable)

yy 100100 5717 TT xx 214 40510 x

17

Uniform Motion problems warm-up (Rate X Time =Distance, distance traveled in time, t, with a uniform rate, r)

1. Dara is traveling at a rate of 40 mph. How far will she travel in 6 hours?

Rate Time Distance

2. How far will she travel in 4 hours?

Rate Time Distance

3. Jim is traveling at 10 mph. How far will he travel in 6 hours?

Rate Time Distance

4. Liz is traveling at 5 miles per hour. How long will it take Liz to travel 35 miles?

Rate Time Distance

4. Liz is traveling at 4 miles per hour. How long will it take Liz to travel 154 miles?

Rate Time Distance

5. If Jimbo travels 45 miles at 10 mph and slows to 30 mph for 5 miles, then how long did it take him to travel

the 75 miles?

Rate Time Distance

Rate Time Distance

18

S-I-D We want to know what x is for the equation to be true

We have to find x =

Simplify Isolate Divide

Simplify xx 17232 xx

10

17

15 2)4(2.03.2 xx

(simplify each side) a) Remove parenthesis -----

Distribute.

b) Remove fractions -----

Multiply both sides by the LCD

c) Remove decimals---

Multiply by powers of 10

d) Combine like terms

Isolate the x's (x’s on one side, non x’s on the other)

(add and subtract)

Divide (Divide by the number in

front of the variable)

Applications:

Try:

1632324 xx 3

12

5

3 xx 5121.2225.0 xx xx 1.253.1

19

Graphing inequalities

<,> ,

Find some solutions to each: 4x x>-5

Graph

Interval notation

Try:

2x x<5 5<x

Graph

Interval notation

Simplify Isolate Divide Fraction--- add/subtract both sides whats next to the variable

Parenthesis----

Linear Inequalities

If you divide or multiply by a negative then

Ex/ -4 ______ 7

413 x 322

313

3

2 kk )4(2)4(35 xxx

20

Phrases

The following four phrases show operations between two things. The create quantities, parenthesis:

The difference of 4 and 3 The sum of 4 and 3 The product of 4 and 3 The quotient of 4 and 3

( 4 - 3) ( 4 + 3) ( 4 · 3) ( 4 3)

The product of 4 plus 3 and 7 minus a number. The product of 4 plus 3 and 7, minus a number

x 734 x 734

Example: four times the sum of five and six.

4 · ( 5 + 6 ) 4 (5 + 6)

TWO SPECIAL CASES

A number subtracted from 3 7 less than a number

X - 3 7 - X

3 - X X - 7

Translate each phrase and solve the equation

1. The product of 3 and a number is 4 more than the number

2. The sum of 7 and 8 3. 4 subtracted from 8 is equal to the product of a number and 3

4. The quotient of 6 and 3 is equal to a number plus 3

5. 8 times the sum of a number and 3 is equal to the same number

6. The difference between six times a number and four times the number is negative fourteen.

21

1) The sum of two numbers is 83. One of the numbers is 11 more than the other. What are the numbers?

a) Define your variables. The sum of two numbers is 83.

One number is= ______________________ then the other number is=_______________.

b) Form an equation.

Equation: One of the numbers is 11 more than the other.

2. The sum of two numbers is eighteen. The total of three times the smaller and twice the larger is forty-four.

Find the two numbers.

a) Define your variables. The sum of two numbers is eighteen.

One number is= ______________________ then the other number is=_______________.


The total of three times the smaller and twice the larger is forty-four.

3) The width of a rectangular garden is one-third its length and its perimeter is 32 m. Find the dimensions of

the garden.

Width=________ Length=___________ Equation: Perimeter = 2length + 2width

4) The fee charged by a ticketing agency for a concert is $5.50 plus $37.50 for each ticket purchased. If your

total charge for tickets is $343, how many tickets are you purchasing?

a) Define your variable. X=___________________________


The fee charged by a ticketing agency for a concert is $5.50 plus $37.50 for each ticket purchased

5) A union charges monthly dues of $4 plus $0.15 for each hour worked during the month. A union member's

dues for March were $29.20. How many hours did the union member work during the month of March?

a)

b)

22

1) Three consecutive positive integers have a sum of 36.

a) Define your variables.

Integer 1is _________________ Integer 2 is__________________ Integer 3 is____________________

b) Form and equation. Sum of 36.

Equation: _______+________+___________=36

2) The sum of three consecutive odd integers is 57. Find the integers.


Integer 1is _________________ Integer 2 is__________________ Integer 3 is____________________

b) Form and equation. Sum of ____.

Equation: _______+________+___________=

3) Find two consecutive even integers such that six times the first integer equals three times the second integer.


Integer 1is _________________ Integer 2 is__________________

b) Form and equation.

such that six times the first integer equals three times the second integer

4) Three times the smallest of three consecutive even integers is two more than twice the largest. Find the

integers.

23

Coin Problems

If you have 25 quarters, how much do you have?

If you have x quarters, how much do you have?

If you have y nickels, how much do you have?

If you have 25 coins and 10 are nickels, then how many coins are not nickels?

If you have 25 coins made up of nickels and quarters and 10 coins are nickels, then how

many quarters do you have?

If you have 25 coins made up of nickels and quarters and X coins are nickels, then how

many quarters do you have?

Nana Nantuambu found some coins while looking under her sofa pillows. There were equal numbers of nickels

and quarters, and twice as many half-dollars as quarters. If she found $2.60 in all, how many of each

denomination of coin did she find?

Amount from nickels Amount from Quarters Amount from half-dollars

John Joslyn has a jar in his office that contains 39 coins. Some are pennies, and the rest are dimes. If the total

value of the coins is $2.64, how many of each denomination does he have?

Amount from pennies Amount from Dimes

24

Lucinda found some stamps while looking under her sofa pillows. There were equal numbers of 5 cent stamps

as 32 cent stamps, and twice as many 15 cent stamps as 32 cent stamps. If she found $4.69 in all, how many of

each type of stamp did she find?

Amount from 5 cent stamps Amount from 32 cent stamps Amount from 15 cent stamps

John Joslyn has a jar in his office that contains 39 stamps. Some are one cent stamps, and the rest are ten cent

stamps. If the total value of the stamps is $2.64, how many of each type of stamp does he have?

Amount from 1 cent stamps Amount from 10c ent stamps

A bank teller cashed a check for $200 using twenty-dollar bills and ten-dollar bills. In all, 12 bills were handed

to the customer. Find the number of twenty-dollar bills and the number of ten-dollar bills.

25

120

(4x-8)

(2x+16)

(2x+10)

(4x-10)

(4x-8)

(2x)

60

110

y x

1) Name the type of angles below:

2) Interior angles of a triangle add up to________________________,

1) Find the missing angles 2) Find the missing angles 3) Find the missing angles

4) Find the missing angles 5) Find x and y

26

Find the missing percentage.

% PER 100 PARTS PER 100 PARTS 100

100

27% 27 PER 100 27 PARTS PER 100 PARTS 10027

10027 = 0.27

25% of is

40% of is

25% of 42 is_________________

33% of 35 is_________________

What is 10% of 70 __________________________

Increase 70 by 10%

Decrease 70 by 10%

A $70 shirt is 10% off The cost of the shirt is___________

Answer the following mentally:

1) Find the given percentage of 50: a) 10% b) 20% c) 60% d) 5% e)1%

2) Increase 50 by: a) 10% b) 20% c) 60% d) 5% e)1%

3) Decrease 50 by: a) 10% b) 20% c) 60% d) 5% e)1%

i)You are looking at shirt that costs $40 and you have a 25% off coupon. How much will you pay before tax?

ii) How much will your total be after tax? (8% tax)

iii) If your credit card bill increased 2% every month and your current bill is $1500, then how much will your

next bill be?

27

SELLING PRICE= COST + MARKUP MARKUP = (MARKUP RATE)(COST)

S = C+M M= R(C)

S = C + _____

EX1/ You purchase a shoe for $4 at a yard sale and want to selling for a 25% profit. What should you sell it

for?

EX2/ You find the matching shoe, but have already sold the other shoe. You paid $5 for the new shoe and sell

it for $7. What is the Markup rate?

1. You purchase a monkey for $5000 and decide to sell it for a 55% markup. What should you sell the monkey

for?

2. You find a diamond ring in the middle of the street, but end up getting a $45 jay walking ticket. If you want

to make a 515% profit on the ring, then what should you sell it for?

3. A 46" LCD TV is marked-up 45% and sold for $2499. What is the cost to the seller?

4.. A snicker bar is purchased for 27 cents and sold for 99 cents. What is the mark up rate?

If you mark an item up 80% of the original price, then what percentage of the original price is the selling

price?

+ = _________% of original price

Selling price =

28

If you have a 20% off coupon for a pair of shoes then:

your discount rate is what percentage________________________

the percentage of the original that you pay is__________________

Discount= (discount rate)%Original price

Sale price = Original price - Discount

S = P - ______

1. a) A shirt you have been looking at is 40% off. If the regular price is $25, then what is the discount for the

shirt?

40% of $25 is_________________

b) What is the Sale price for the shirt?

2. You have a 40% off coupon for your entire purchase. If the regular prices for your items are $4, $5, and $7,

then what is your discount?

a) Sum of the items is___________

b) What is the Sale price for all of your items?

3. A bedroom set that normally sells for $1450 is on sale for 10% off. Find the price after the discount?

4. A wholesaler is selling slugs and offers a 10% discount if you buy 100 to 200 and a discount of 20% if you

buy more than 200. If the price of 1 slug is 10 cents, then find the price after the discount for 350 slugs.

5. A shirt is on sale with a discount of 20%. If the sale price is $35.95, what was the original price?

If you have a 20% off coupon then what percentage of the original price are you paying for?

29

Simple Interest= (simple interest rate)%TotalYears

1. How much simple interest is earned on $5,000 if invested for 1 year at 5%?

Interest rate Principle Years Simple Interest

=

2. a) How much simple interest is earned on $2,000 if invested for 1 year at 6%?

Interest rate Principle Years Simple Interest

=

b) How much money is in the account?

$4,000 $3,000 __________ $5,000 _____ $11,000

X _______ $60,000 ______ x $4,000

So if I invest $40,000 into two accounts one at 4% and the other at 6%, then I invested in terms of x:

______________into the 4% and _______________into the 6%.

30

Investment Problems I=Pr

Melissa Wright won $60,000 on a slot machine in Las Vegas. She invested part at 2% simple interest and the

rest at 3%. In one year she earned a total of $1,600 in interest. How much was invested at each rate?

Principle Rate = Interest

1st account

2nd account

Michael Pellissier invested some money at 4.5% simple interest and $1,000 less than twice that amount at 3 %.

His total annual income from the interest was $1020. How much was invested at each rate?


1st account

2nd account

Dee invested 40% of his money at an annual simple interest rate of 9%. The remainder of her money was

place in an annuity earning 7%. The total interest earned was $2496. Find the total amount invested.


1st account

2nd account

Jimbo invested 30% of his cash in a 6% simple interest fund, 30% in stocks earning 7%, and the remainder in

an account earning 8%. The total interest earned for the year was $5437.50. Find the total amount invested.

31

1) A solution of alcohol and water is 80% alcohol. If you have 40 liters of the solution, then how many liters are alcohol ?

2) A solution of alcohol and water is 80% alcohol. If you have x liters of the solution, then how many liters are alcohol ?

3) If you mix 4 liters of a solution with 5 liters of a solution, then you have how many liters of the solution?

4) If you mix 4 liters of a solution with x liters of a solution, then you have how many liters of the solution?

Mixture Problems

How many liters of a 14% alcohol solution must be mixed with 20L of a 50% solution to get a 30% solution?

How much pure dye must be added to 4 gal of 25% dye solution to increase the solution to 40%?

32

To make a flour mix, a miller combined soybeans that cost $8.50 per bushel with wheat that cost $4.50per bushel. How many bushels

of each were used to make a mixture of 800 bushels that cost $5.50 per bushel.

$ + $ = $mixture

A hair dye is made by blending 7% hydrogen peroxide solution and 4% hydrogen peroxide solution. How

many milliliters of each are used to make a 300-milliliter solution that is 5% hydrogen peroxide?

How many ounces of pure chocolate must be added to 150 oz of chocolate topping that is 50% chocolate to

make a topping that is 75% chocolate?

A bar tender combines 12 oz of 20% alcohol with and 8 oz bottle of water. What is the percent concentration of

alcohol in the new bottle?

How many pounds of chicken feed that is 50% corn must be mixed with 400lb of feed that is 80% corn to make

a chicken feed that is 75% corn?

33

Uniform Motion problems warm-up (Rate X Time =Distance, distance traveled in time, t, with a uniform rate, r)

One person:

1. Dara is traveling at a rate of 40 mph. How far will she travel in 6 hours?

Rate Time Distance

2. How far will she travel in T hours?

Rate Time Distance

3. Jim is traveling at x mph. How far will he travel in 6 hours?

Rate Time Distance

Two people:

1. Liz is traveling at x miles per hour. Leann is traveling at twice the speed of Liz.

How far will Liz travel in 3 hours?

Rate Time Distance

How far will Leann travel in 3 hours?

Rate Time Distance

2. James is traveling at 22 miles per hour. Silvia is traveling at 33 miles per hour. James has been traveling for

t hours and Silvia left 30 minutes before James.

How far has James traveled?

Rate Time Distance

How far has Silvia traveled?

Rate Time Distance

34

TRDsnail TRDsalt

TDsnail 4 TDsalt 2

minutes2

126

1224

12

T

T

TT

DD saltsnail

TRD 1 TRD 2 TRD 1 TRD 2

TRD 1

TRD 2

TRDS

TRDJ

Uniform Motion (d=rt) 1) It’s high noon (12:00) on a hot summer day. A mildly mannered Snail spots a salt shaker known as Sergeant

Salt, and in a near by phone booth becomes Super Snail. Super Snail and Sergeant Salt are going to rumble.

The Snail and Salt shaker are 12 inches apart. If Super Snail travels at 4 inches per minute and Sergeant Salt

travels at 2 inches per minute, then when will they rumble?

So at 12:02 pm they will rumble.

2) Little Jimmy and Little Sally are going to race. Little Jimmy is focused on an ant walking in front of him

when the race starts. It takes Jimmy 3 seconds to realize the race has started. Jimmy travels at 2 ft/sec. Sally

twisted her ankle, so she can only travel at 1 ft/sec. When Jimmy starts the race, how long will it take Jimmy to

catch Sally?

DistanceTotal21 DD DistanceTotal21 DD 21 DD

3) A train leaves Kansas City, Kansas, and travels west at 85 km/hr. Another train leaves at the same time and

travels east at 95 km/hr. How long will it take before they are 315 km apart?

4) When Dewayne drives his car to work, the trip takes 30 min. When he rides the bus, it takes 45 min. The

average speed of the bus is 12 mph less than his speed when driving. Find the distance he travels to work.

rtd train 1 rtd train 2

35

5) Two cyclists start at 1:00 pm from opposite ends of a 54-mile race course. The average rate of speed of the

first cyclist is 17 mph, and the average rate of speed of the second cyclist is 19 mph. At what time will the two

cyclists meet?

Picture :

6) A helicopter traveling 120 mph overtakes a speeding car traveling 90 mph. The car had a 0.5 hour head

start. If they started from the same point, how long did it take the helicopter to overtake the car?

Picture:

ii) Question number 2. How far did the helicopter travel before overtaking the car?

Last one.

6) Two planes start from the same point and fly in opposite directions. The first plane is flying 50 mph slower

than the second plane. In 2.5 hr, the planes are 1400 mi apart. Find the rate of each plane.

Picture:

36

Inequality Equations

List some numbers that are less than 10 x is less than 10

List some numbers that are greater than 10 x is greater than 10

List some numbers that are at least 10 x is at least 10

10,

List some numbers that are at most 10 x is at most 10

10,

List some numbers that are a minimum of 10 x is a minimum of 10

10,

List some numbers that are a maximum of 10 x is a maximum of 10

10,

1. A health official recommends a maximum cholesterol level of 120 units. A patient has a cholesterol level of

275. By how many units must this patient's cholesterol level be reduced to satisfy the recommended maximum

level?

2. A professor scores all tests with a maximum of 100 points. To earn an A in this course, a student must have

an average of at least 92 on four tests. One student's grades on the first three tests were 89, 86, and 90. Can

this student earn an A grade?

3. A car sales representative receives a commission that is the greater of $250 or 8% of the selling price of a

car. What dollar amounts in the sale price of a car will make the commission offer more attractive that the $250

flat fee?

4. A residential water bill is based on a flat fee of $10 plus a charge of $.75 for each 1000gal of water used.

Find the number of gallons of water a family can use and have a monthly water bill that is less than $55.

37

217 yx10 yx

3y

x y x y

5x

RECTANGULAR COORDINATE SYSTEM

Plot (3,-2) , (4,0) , (0,0) , (-5,0) (0,6)

(-3,2) , (-4,0) , (0,1) , (1,0) (0,-6)

Is (3,4) a solutions to the equation x + y = 7 ?____________________

What values of x and y make the equation true?

The coordinates that make the following equations true are called solutions to the equation.

x yx y

38

23 xy xy 2

X- and Y- intercepts

2054 yx X-intercept Y-intercept Let y=0 Let x=0




32

1 xy X-intercept Y-intercept

Let y=0 Let x=0

Find the x and y-intercept and graph

39

m= m=

m= m=

m=Slope = ______________=_______________=_______________

1. a) Graph the points (4,2) and (-3,0) b) Find the slope using the formula

Find the slope by counting . __

Slopes of Perpendicular and Parallel Lines 1. One line passes through (–1, – 4) and (2, –10). Find the slope of a parallel line.

Slope = _______

2. One line passes through (2, – 4) and (2, 4). Find the slope of a perpendicular line.

Slope = _______

40

y=mx+b

1. Graph the following and label 2 points

a) 32

3 xy b) xy 2 c) 4y d) 2x e) 632 yx

.

a) 12 xy b) xy4

5 c) 2x d) 1053 yx e) 7y

41

Then verbally interpret.

hours

milesm

every

Distance traveled increase_____________ every _________

What happens to y= 4x + 3, if you change it to y= 4x + 7 ?

What happens to y= 4x + 3, if you change it to y= - 4x + 3 ?

42

bmxy

Finding the equation of the line. 1st find the slope 2nd find the y-intercept

1. Given a point (3,4) on the line and a slope of -3. Find the equation of the line.

Or

2. Given two points on the line (3,2) and (4,3). Find the equation of the line.

Or

3. Write the equation of the line parallel to y = –2x + 2 with a y-intercept of ( 0,4)

y = ______x + ______

Write the equation of the line perpendicular to y = 3x + 4 with a y-intercept of ( 0,4)

y = ______x + ______

4. Given a point (-2,0) and the line is perpendicular to y = 4x + 3.

Or

5. Given a point (-2,0) and the line is parallel to y = 4x + 3.

Or

43

44

FUNCTIONS

Ordered pair

Relation

Function

Determine if the following are functions:

3,2,1,2,0,1 3,3,1,2,0,1

Domain: Range:

Determine the domain and range for the following:

3,2,1,2,0,1 3,3,1,2,0,1

)(xf means the value of the function f at x . Be careful )(xf doesn’t mean f times x .

Let’s start with a function 42)( 2 xxxf

)1(f

)3(f *Always place the substituted number in

parenthesis.

)5(f

f ( ) =

)(f

)(kf

3,5,1,4,3,2

3,5,1,4,3,2

45

yxf )( 1,2, yx

52 xy 52)( xxf

1522)2( f

Change the f(x) to y and graph.

a) 12)( xxf b) xxf4

5)( c) 2x d) 7)( xf

If 15)( xxf , and the domain of x is 5,4,2 , then what is the range of the function?

I i) Find two points, then calculate the slope. ii) Verbally interpret slope and y-intercept.

a) b)

2. The average price of a Porsche 997 is given by the equation 000,904000 xP , where x is the number of

years.

a) Write the equation in functional notation.

b) Find C(50), and tell me what your answer means.

46

Find and plot some ordered pairs (x,y) that make the following inequalities true.

10 yx 2 xy

, -------------------Dotted line , Solid line

Graph:

3x xy 2 32 xy 632 yx

02 x 02 x 02 yx 023 xy

035 yx 035 yx 02 y 02 y

47

Systems of Linear Equations Graph and find the solutions to the systems of equations:

1.

3

932

x

yx 2.

23

13

xy

xy 3.

42

42

xy

xy

In exercises 1 – 8, solve the system of equations by GRAPHING.

1) 4 7 2) 2 5 2 3) 2 5 4) 3 7 2

2 3 3 8 2 5 4

x y x y x y x y

x y x y x y

2 3 14

5) 3 4 6) 4 3 7) 3 2 8) 5 2

2 7 8

x y

x y x y x y x y

x y x y

6 5 7 y x y x

48

yx

yx

2

423

-=

=+

21312

74

yx

yx

52

234

yx

yx

SUBSTITUTION METHOD

X Y

Substitute the X

yx 2-= 423 =+ yx

( ) 423 =+ y

,

,

,

49

In exercises 1 – 8, solve the system of equations using the substitution method.

1) 4 7 2) 2 5 2 3) 2 5 4) 3 7 2

2 3 3 8 2 5 4

x y x y x y x y

x y x y x y

2 3 14

5) 3 4 6) 4 3 7) 3 2 8) 5 2

2 7 8

x y

x y x y x y x y

x y x y

6 5 7 y x y x

50

52

234

yx

yx

21912

534

yx

yx

ELIMINATION/ADDITION METHOD 242

723

yx

yx

Substitute:

Multiply: 242 yx

242

723

yx

yx

,

Substitute:

Multiply:

,

51

In exercises 1 – 8, solve the system of equ ations by the correct method.

1) 4 7 2) 2 5 2 3) 2 5 4) 3 7 2

2 3 3 8 2 5 4

x y x y x y x y

x y x y x y

2 3 14

5) 3 4 6) 4 3 7) 3 2 8) 5 2

2 7 8

x y

x y x y x y x y

x y x y

6 5 7 y x y x

Now try: xyx

yyx

93

133

52

D = R * T

D = R * T

D = R * T

D = R * T

D = R * T

D = R * T

Uniform Motion (d=rt) When Dewayne drives his boat to work up stream , the trip takes 45 min. When he comes home downstream, it

takes 30 min. The distance to work is 90 miles. Find the speed of the stream.

X= Speed of the boat in calm water Y= Speed of the stream.

upstream

downstream

A jet plane flying with the wind went 2100 mi in 4 hr. Against the wind, the plane could fly only 1760 mi in

the same amount of time. Find the rate of the plane in calm air and the rate of the wind.

X=________________________________ Y=_______________________________

against wind

with wind

A rowing team rowing with the current went 18 km in 2 hr. Against the current, the team

rowed 12 km in the same amount of time. Find the rate of the team in calm water and the rate

of the current.

X=________________________________ Y=_______________________________

against current

with current

53

1. If one shirt costs $10, then what is the total cost for 5 shirts?

2. If one dog costs $25, then what is the total cost for 4 dogs?

3. What would be the total cost for 6 shirts and 2 dogs? Show your work.

4. What would be the total cost for x shirts and y dogs? Show your work.

5. If you purchased 6 cats for x dollars, then what is your cost?

6. If you purchased 5 fish for y dollars, then what is your cost?

7. If you purchased 2 cats for x dollars each, and 7 fish for y dollars each, then what is your total cost?

8. If you have x $5 bills, then how much do you have?

9. If you have y $10 bills, then how much do you have?

10. If you have x $5 bills and y $10 bills, then what is the total value?

1. The manager of a discount clothing store received two shipments of fall clothing. The cost of the first

shipment, which contained 10 identical sweaters and 20 identical jackets, was $800. The second shipment, at

the same prices, contained 5 of the same sweaters and 15 of the same jackets. The cost of the second shipment

was $550. Find the cost of one jacket.

Describe: X=_______________________, Y=____________________________

Equation 1: (first shipment)

Equation 2: (second shipment)

2. A computer online service charges one hourly rate for regular use but a higher hourly rate for designated

"premium" areas. One customer was charged $14 after spending 2 hours in premium areas and 9 hours in

regular areas. Another customer spent 3 hours in premium areas and 6 hours in regular areas and was charged

$13.50. What is the service charge per hour for regular and premium areas?

Describe: X=_______________________, Y=____________________________

Equation 1:

Equation 2:

3. Two coin banks contain only nickels and dimes. The total value of the coins in the first bank is $3. In the

second bank, there are 4 more nickels than in the first bank and one-half as many dimes. The total value of the

coins in the second bank is $2. Find the number of nickels and the number of dimes in the first bank.

Describe: X=_______________________, Y=____________________________

Equation 1: + = 300

Equation 2: + = 200

54

Polynomials -------

xx

x

xx

xx

3

1

3

12

32

1

34

23

Degree of a Polynomial

xx

x

xx

xx

3

1

3

12

32

1

34

23

Identify the following polynomials:

xx

x

xxx

xx

3

1

23

2

32

1

234

423

Descending order

723 51432 xxxx

Leading Term

Leading Coefficient

The constant (no variable)

55

+

1 + 3 4

The number in front tells you how many you have.

3 + 2 + 2 8x - 5x² + 6x²

2542537 23235 xxxxxx

2542537 23235 xxxxxx

536186 233 xxxx 2568 35 xxx

2

2

1 335 xxx 5.3335.062.005.003.1 32 xxxx

(3 -2 +4 -5 )-( - +4 )

2542537 2332335 yxxyxxyxyyx

56

Multiplying Monomials Expand and find the answer:

32 23 xx 725 xx 225 32 yxyx 23 xxy

What operation is happening between the exponents?_______________________

Warm up:

22 32 42 52 Do you see a pattern?____________

22 32 42 Do you see a pattern?____________

Expand and find the answer:

23x 234 xy 23yx 23x 232x

What operation is happening between the exponents?_______________________

Now all together:

6327 5672 aaaa 233322 abaaba

57

Monomial (polynomial)

232 223 xxx 7825 xxx 32 282 xxx

BinomialBinomial

5234 xx 5234 xx 5234 xx

The actual multiplication F O I L

5234 xx 1523 xx 23yx

BinomialTrinomial

52334 2 xxx YxYxYx 523 2

58

a) Find: i) perimeter ii) area

13 3 y

25 3 y

b) Find: Area

12 y

42 y

Simplify:

62322

bbbaba

59

RULES EXPONENT

exponentbase

Warm up:

22 32 42 52 Do you see a pattern?____________

22 32 42 Do you see a pattern?____________

225 32 yxyx 23 xxy 23x 232x

1. pmpm aaa ex/ 32 23 xx

2. pmpm aa ex/ 23x

3. mmmbaab

How many bases are their?

m

mm

b

a

b

a

a)

2

2

2

2

2

3

2

3

zx

yx

xz

yx

Ex/ 2

6332

3 4222

y

x

y

x

y

x

y

x

b) 22 33 xx

2

3

y

x

223

y

xz

y

x

y

xz2

23 3

2

2

3x

y

x

60

4a

4. pm

p

m

aa

a

ex/ x

xxx

x

x3

ex/ xyyyyy

xxyyy

xy

yx5

32

ex/ _____0

3

3

xx

x

Now for the faster method:

4

23

4

2

xy

yx

213

2

6

8

yx

yx

czxy

zyx84

323

4

2

5. m

m

aa

1

p

pa

a

1 What does the negative exponent do?__________________

Ex/

43

22

yx

yx Ex/

85

25

8

4

yx

yx

83

56

3

9

yx

yx

83

5

8

4

yx

x

a

b

b

a

1

Why did the inside flip?_____________________

33

a

b

b

a

1

5

x

35

x

1225

x

x

24

4

432 333 12 42

12

3

5

3

2

3

312 23

baab

2

85

25

8

4

yx

yx

61

Combine Give everyone the exponent Kill Travel

342 bbaa 323 ba

6

2

6

4

a

a

1

44

bc

a

22423 yyxx 215 xy 34

52

12

8

zxy

yzx

332

xy

x

42

2

6

3

qp

qpq

5

432

6

12

k

kk

62

SCIENTIFIC NOTATION

3,654,000,000,000,000,000 0.000000000123

1) 32300000000 10.

2) 2588800000000000 10.

3) 56575 10.

4) 0.000000000562 10.

5) 0.789 10.

6) 0.000000589 10.

7) 56.8

8) 99999

9) 123.55

10) 0.0000000005065

Scientific Notation to Decimal form

1) 61015.3

2) 61015.3

3) 51035.9

4) 1510789.1

5) 4109

6) 6-103

7) 361065555.9

63

26 102.81026.4 XX

Answer in standard notation:________________________

57 108.1102.4 XX


3

9

105.2

105.7

X

X


2

8

105.2

106

X

X


Dividing by Monomials

4 -2 -5

2

x

xx

2

423 2

xy

yxyyx

4

423 42

64

4q

_________

___________

2344 2 qqq

4q

4

3242

q

qq

q

q 2

q

q8

90933

Steps:

1)

2)

3)

4)

Answer:

4

102

a

aa

32

211116 23

x

xxx

4

572 2

q

qq 3

923 3

x

xx

1

17

x

x

65

How to find the GCF

70 28 42 135

7 10 4 7 7 6 5 27

2 5 2 2 2 3 3 9

3 3

7 x 2 x 5 2 x 2 x 7

GCF = 1472

7 x 2 x 3 5 x 3 x 3 x 3

GCF =

1. Divide out the numbers they have in common.

18 24

GCF=

100 40

GCF=

Find the GCF.

1) 18 8 2) 36 64 3) 18 45 36

66

FACTORING

FACTORING OUT THE GCF

Factoring23123

ngDistributi12343

xx

xx

EX 1/ Factor 24 + 36x you can do it in parts

24 + 36x

EX 2/ Factor 36x+24x

36x+24x

EX 2/ Factor yc36x+3y24x

yc36x+3y24x

1. Simplify 4323 2 xxx 2. Factor xxx 1296 23

3. Simplify 243 2 xx 4. Factor 23 612 xx

What is the greatest common factor?___________


Do the two terms have an x in common, how many?__________


Do the two terms have an x in common, how many?__________

Do the two terms have any other common factors?__________

67

5. Simplify yxzzxyxyzxyz 32 6. Factor zxyyzxzyxzyx 222242322

7. Factor 3248 2 cc 8. Factor 3223 123640 abcaba

9. Factor 528525 xxx 10. Factor xxx 5857

11)Factor xxx 315132 12) Factor 457453 xxx

4 TERMS

Box Method Factor by Grouping

qymyqxmx 33 qymyqxmx 33

14416 23 mmm 14416 23 mmm

`

68

Factoring Trinomials Part I

25 XX

________________2_______ XX

The last two numbers multiply to give you the third term.

The last two numbers add to give you the middle term

1282 xx ______________ XX

1) What two numbers when multiplied give you the last number (find the factors of 12)

Test numbers :

____·____ ____·____ ____·____

2) What same two numbers when added give you the middle number (add the factors of 12)

____+____ ____+____ ____+____

3) 1282 xx ______________ XX

4) Check it

26 Xx

c-bx+xor c-bx-x 22 then one of the numbers is -,

c+bx-x2 then both of the numbers are negative.

cbxx2 then both of the numbers are positive.

69

1. Fill in the blanks.

(a) To factor 2x + 6x + 8 , you need two numbers that add up to_______ and have a product of_________ .

Then 2x + 6x + 8 = ( )( )

(b) To factor 2x - 6x + 8 , you need two numbers that add up to_______ and have a product of_________ .

Then 2x - 6x + 8 = ( )( )

(c) To factor 2x - 2x - 3 , you need two numbers that add up to_______ and have a product of_________ .

Then 2x - 2x - 3 = ( )( )

d) To factor2x + 2x - 3 , you need two numbers that add up to_______ and have a product of_________ .

Then 2x + 2x - 3 =

2. Factor each trinomial.

In each case, check by multiplying your answer. This allows you to check to see if your answer is correct.

example: b2 + b - 30 = (b - 5)(b + 6)

check: (b - 5)(b + 6) = b2 + b - 30 , so the answer is correct.

No check=No credit

(a) 2x + 7x + 10 (b) 15+

2x - 8x (c) 64202 xx

(d) 22 44 yxyx (e)

22 1716 yxyx (f) xyyxyx 10122 23

g) 32 24 yy h) 15 2x - 12xy + 25xy - 20 2y

i) 2m + 6 m + 27 j) 28210 xx

70

Factoring Trinomials Part II Multiply the following:

752 xx 1753 xx 4343 xx 55 xx

FACTORING TRINOMIALS part 2 30116 2 xx

Something new, so a little different

What two numbers when multiplied give you 6(-30)=____

30116 2 xx

What same two numbers when added give you the middle

number____

Setting the problem up 1) What two numbers when multiplied give 2) What same two numbers when added you

you the number -180 give you the middle number -8

Test numbers

____·____ ____and ____ ____+____

____·____ ____and ____ ____+____

____·____ ____and ____ ____+____

____·____ ____and ____ ____+____

____·____ ____and ____ ____+____

____·____ ____and ____ ____+____

3) So and 30116 2 xx

So we change -8x into

30____________6 2 x

Now we need to factor it

The Box Method or Factor by Grouping

1. Group

(Don’t forget about the negative)

2. Factor each group

71

821849 xx

5823 xx 6122 xx 1526 xx

6725 xx 834235 yy

41529 xx 91524 xx 63028 xx Hint: common factor

181523 xx xx 281628 4356 xxxx Hint: descending order

72

2-TERMS

bababa 22

222 2 bababa

92 x

24 y

42 182 xx

8124x

164 y

2231 xx

962 xx

22 42849 yxyx

73

18A

Look for common FACTORS.

Factor 36x+24x

Factor x224x-36x-

2-TERMS bababa 22 216 x

2233 babababa 364 x

2233 babababa 3327 yx

3-TERMS 432 xx What two numbers when multiplied give you the last number (c)

______________ XX and when added gives you the middle number (b).

12112 2 xx

4- TERMS Box Method Factor by Grouping

qymyqxmx 33 qymyqxmx 33

74

Solving

0123 xx What do you think the answers are?____________

Solving using factoring

562 xx 21019 hh

082 2 a 2224 xx

05136 23 xxx 412 x

2712 xx 5323 yy

75

Rational Expressions

The following are rational expressions. What can x not equal?

2

5

x

x

12

3

xx

x

65

22 xx

FACTOR KILL Monomials

zyx

yx22

5

2

6

zyyxx

yxxxxx

2

6

yz

xxx

1

3

yz

x33

Try this one:

c

dc

15

5 53

Trinomials

( )( )

( )2+

6+2+

x

xx

1

6+x 6+x

Try this one:

16-6

82 xx

x

+

+

( )( )

( )2

22-

+

+

x

xx

1

2-x 2x

Try this one:

x

x

3

9-2

2+

12+8+2

x

XX

2

4-2

+x

x

76

77

LCM

LCM Least Common Multiple

LCM is the smallest multiple that two or more numbers share.

2 3 4 5 6

4 8 12 16 20 24

10 20 30 40 50 60

Find the LCM : each factor must be in the LCM and the largest exponent of each factor must be on that factor..

1. Denominators: 3 zyx 32, 9 yzx5

2. Denominators: 216xy , xz4

LCM: LCM:

3. Denominators: yxx , 32 yxx 4. Denominators: 234

xx , 223 xx

LCM: LCM:

5. Denominators: 323 xx , 3253 xx

LCM:

6. Denominators: 24 2 xx , 226 xxxy

LCM:

6. Denominators: 94 2 x , 932 2 xx

LCM:

78

Find the LCM/LCD

1. 26ab , 318ab 2. 2410 yx , 3315 yx

2. 37 xx , 27x 3. 1x , 1x

3. 452 xx , 2832 xx 4.

3

2

6 2

x

x

xx

5. 2

7

2;

7

3

yy 6.

2;

2

32 x

x

xx

x

7. 20

3;

4;

5

32 xxx

x

x

x

79

Rational Expressions: 32

2

9

3222

xx

x

x

x

1) Factor denominators

13

2

33

32

xx

x

xx

x

2) LCD____________________

3) Multiply top and bottom

13

2

33

32

xx

x

xx

x

4) Add or subtract

5) Reduce if possible

Rational Expressions: 3

52

3

152

yyyy


2) LCD____________________

3) Multiply top and bottom

3

5

2

)3(

15

y

y

yy

4) Add or subtract

5) Reduce if possible

80

1) xyyx 6

11

12

52

2) xx

1

3

3

3) 65

72

65

1322

xx

x

xx

x

4) 2

1

2

3

tt

5) 1

4

1

2

1

42

xxx

6)

42

5

2

32

pp

7)

2222 169

2

456

56

yx

yx

yxyx

yx

81

Complex Fractions.

1)

5

57

x

x

x

2)

7

43

y

y

3)

11

11

a

a

4)

8

1

4

24

xx

5)

xy

xy

yx

35

925 22

6)

2

2

344

1252

xx

xx

7)

1

21

2

11

x

x

82

Rational Equations: 63

1

9

11

2

4

mm


)2(3

1

9

11

2

4

mm

2) LCD____________________ x ____________

3) Multiply both sides (all groupings) by the LCD

)2(3

1

9

11

2

4

mm

4. Solve the resulting equation

Rational Equations: 2

1

4

2

2

32

kkk


2) LCD____________________ x ____________

3) Multiply both sides (all groupings) by the LCD

4. Solve the resulting equation

83

1) 224

x

x

2)3

15

2

85

x

x

x

x

3)26

2512

32

14

x

x

x

x

4) 152

48

5

7

3

62

wwww

5) 3

42

9

252

9

1452

2

2

x

x

x

xx

x

x

6) Which values for x would not be possible for the

following equations? (hint: denominator cannot =?)

a) xx

42

3 b)

3

54

2

3

xx

c) 12

432

xx d)

23

5

2

42

xx

x

x

7) Go back and check numbers 1-5 for possible no

solutions or not possible

84

63

1

9

11

2

4

mm

2),2(9 mmLCD

)2(3

1

9

11

2

4

mm

)2(9

)2(9)2(9

)2(3

1

9

11

2

4

m

mm

mm

3)2(1136 m

1m

32

2

9

3222

xx

x

x

x

13

2

33

32

xx

x

xx

x

313

62

313

32 22

xxx

xx

xxx

xx

313 xxxLCD

3

3

13

2

33

32

1

1

x

x

xx

x

xx

x

x

x

313

354 2

xxx

xx

8

1

4

24

xx

xxx

xx

x

88

18

4

82

84

xx

xx

88

1

4

82

4

12

1216

xx

x

xx

x

22

1632

x

16

xLCD 8

)2)(2(9

)2(3

)2)(2(9,23

52

3

532

xxyx

xxy

xxxyxyx

Rationals

85

86

Simplify Isolate Divide Fraction--- add/subtract both sides whats next to the variable

Parenthesis----

Solving for a given variable (formulas)

a) DPS , for D b) yarr 32 , for r c) bBhA 2

1, for b d)

2d

kL , for k

tPA Pr , for P 329

5 FC , for F CRP , for t

87

Work Problems

Distance = Rate(time)

Work completed = Rate(time) W=R*T R=

Rate= the speed at which a person/thing can perform, travel, or do something= it complete to Time

completed Work.

Example: If Jimbo can mow one lawn in 25 minutes. Then his rate would be:

minutes 25

lawn 1 = minute /lawns

25

1

Find the rate:

1. Johnny can was 6 dogs in 2 hours.

2. Tommy can write 40 words in 2 minutes.

3. A clerk can stock a shelf in 20 minutes.

4. A clerk can stock a shelf in 30 minutes.

5. Sally can fill 4 bottles every t hours.

6. Jimmy can squish 1 ant every t minutes.

7. Working together Jimmy and Sally can mow 1 lawn in t minutes.

Finding the work completed: Work completed = Rate(time)

1. Johnny can wash 6 dogs in 2 hours. How many dogs can he wash in 3 hours?

2. Tommy can write 40 words in 2 minutes. How many words can he write in 5 minutes?

3. A clerk can stock a shelf in 20 minutes. How many shelves can he stock in 1 minute?

4. A clerk can stock a shelf in 30 minutes. How many shelves can he stock in 1 minute?

So the work completed after one hour is the same as the rate itself. Why?

Ex1/ One grocery clerk can stock a shelf in 20 min. A second clerk requires 30 min to stock the same shelf.

How long would it take to stock the shelf if the two clerks worked together?

a) Clerk 1's rate: Clerk 2's rate:

b) Rate together = Clerk 1's rate + Clerk 2's rate

= +

88

1) Jimmy can mow a lawn in 75 minutes. Sally can mow a lawn in 50 minutes. How long would it take them

to mow the lawn together?

a) Jimmy's rate: Sally's rate:

b) Rate together = +

2. A small air conditioner can cool a room in 60 minutes. A larger air conditioner can cool the room in 40

minutes. How long would it take to cool the room with both air conditioners working together?

3. Jimmy can stock three times as many shelves in the same time it takes Tommy to stock one shelf. If they

work together, then it takes them two hours. How fast can Jimmy stock one shelf and how fast can Tommy

stock one shelf?

Uniform Motion part II

1. Dara is traveling at a rate of 40 mph. How far will she travel in x hours?

Rate Time Distance

2. If Jimbo travels 45 miles at 10 mph and 50 miles at 25 mph, then find the total time traveled?

Rate Time Distance

89

Solve for T

3. The president of a company traveled 1800 mi by jet and 300mi on a prop plane. The rate of the jet was four

times the rate of the prop plane. The entire trip took 5 hours.

TRD T

Distance Rate T

Jet

Prop Plane

Time traveled by Jet=________________________ Time traveled by Prop Plane:________________

Time traveled by Jet + Time traveled by Prop Plane = Total time traveled

A engineer traveled 165 mi by car and then an additional 660 mi by plane. The rate of the plane was four times

the rate of the car. The total trip took 6 hours. Find the rate of the car.

Distance Rate T

Time traveled by Car=________________________ Time traveled by Plane:________________

Time traveled by Car + Time traveled by Plane = Total time traveled

90

1. On a recent trip, a trucker traveled 330 mi at a constant rate. Because of road conditions, the trucker then

reduced the speed by 25 mph. An additional 30 mi was traveled at the reduced rate. The entire trip took 7

hours. Find the two speeds.

Distance Rate T

2. As part of a conditioning, a jogger ran 8 mi in the same amount of time as it took a cyclist to ride 20 mi. The

rate of the cyclist was 12 mph faster than the rate of the jogger. Find the rate of the jogger and the rate of the

cyclist.

Distance Rate T

Hint: the times are equal.

91

SQUARE TO SQUARE ROOT

24

39

416

36

49

64

81

100

121

4

Square roots want the root of the squares.

64 64.0

25

64

4

1

15816 81

64

9

25

25

22

33

44

55

66

77

88

99

1010

1111

1212

92

Radical Land

32 232x 3232 yx 113232 zyx

13481 yx 10145 yx 2281 x 962 xx

236x 27x 28 t 224 yx

* ____2 x ____72 ____92 ____7 2 x ____72x

Adding and Subtracting Radicals: 1 + 3 4

3532 2 square root of threes + 5 square root of threes = 7 square root of threes = _______________

252628 3274379 32 27163125 baabbab

93

Simplify:

1 a) 9392 b) 816

2. a) 459392 b) 32430

3. a) 2754 b) 1085

Multiply: _____________3372

How do you multiply radicals?__________________________________________________________-

Multiply the Radicals and simplify:

217 425247 94 93 xx 9227 99 yxyx

Multiplying Radicals part II: 2xxx 77 aa 88

226432 43 xx 7372 xx 2232

94

Dividing Radicals:

Check list: 6

225

b

a

10

270

1)

2)

3)

a7

7

ba372

9

Multiply:

1) 33 xx 2. 33 xx 3. xx 234234

Conjugate of 23 is 23 Conjugate of 74 x is 74 x

What happens to the radicals when you multiply an expression by its conjugate?_________________________

Rationalizing the Denominator:

74

3

95

zx

z

25

57

Warm up:

1) 23 2) 24 3) 29 4) 23

What does 2

7

What does 2

x Explain.________________________________

What do you need to do to both sides of the equation to remove the ?______

Try to solve the following and then check your answers:

5) 5x 6) 32 x 7) 312 x

96

Try to finish the following:

8)

________

3

3

22

2

x

x

9)

________________

3232

322

xx

x

Was problem 8 or problem 9 faster?__________________

Which answer is void of radicals?___________________

Try to Solve the following:

10) 53 x 11) 32 x 12) 32 x

What must you do before you square both sides?_________________________

True or false a) 22164 xx Why?______________________________

b) 22164 xx Why?______________________________

c) If x=9, then 3x Why?__________________________

97

ISOLATE the radical and square both sides

156 x 053 x 524 x

Check your answer Check your answers Check your answers

Try these:

45410 x 2435 x xx 112

98

42 x 812 x 92 x

Review: Solving using factoring

562 xx 412 x 082 2 a

xxx 5136 23 17323 xxx

Square Root Property.

x= or x= x= or x= x= or x=

Solve: 1442 x 82 x 722 x 542 x

Isolate the Square and then take the square root.

93 2 x 063 2 t 023 2 t

56172 x 2 241

2x 12

2

12

2

z

99

What # completes the square?______________

Completing the Square Warm Up:

1. 2222

xxx 2. 23x

44

422

2

2

xx

xxx 962 xx

3. 25x 4.

2

12

1

x

25102 xx 144

1

6

12 xx

1. Look at the previous and fill in the blanks without factoring:

a) 2___x b) 2___x c) 2___x

442 xx 1682 xx 962 xx

d) 25102 xx e) 9

1

3

22 xx f) 4

12 xx

2___x 2___x 2___x

g) ____82 xx h) ___102 xx i) ___52 xx

2___x 2___x 2___x

So

2____

_____82

x

xx

100

Need to find out what # completes the square

0362 xx

362 xx

2

1

2

362 xx

Try: 011102 xx

0652 2 xx

2

1

2

Try: 362 2 xx

101

a

acbbx

2

42

The negative boy

Couldn’t decide

Whether to go to the radical party

Where the square bouncer

Removed 4 alcoholic chicks

And it was all over at 2 am.

Quadratic Formula

02 cbxax

0432 2 xx

Solve. 362 kk

Solve. 02320 2 yy

Solve. 24715 xxxx

Solve for x using completing the square: 02 cbxax

102

43

43

43

43

9596

932

32

6

2

56

056

2

2

2

2

2

2

2

x

x

x

x

xx

b

b

xx

xx

4

65

4

5

4

65

4

5

16

65

4

5

16

65

4

5

16

25

2

5

16

25

2

5

16

25

4

5

2

4

5

2

5

2

1

2

1

2

2

5

2

5

02

5

2

5

0552

2

2

2

22

2

2

2

x

x

x

x

xx

b

bb

xx

x

xx

22

8

8

2

2

x

x

x

51

51

51

51

2

2

x

x

x

x

1,3

21,21

21

21

41

41

2

2

x

x

x

x

x

x

3

151

3

151

3

51

3

52

513

2

2

2

x

x

x

x

x

a

acbbx

2

42

9

3

3

22

x

x

x

8

2

2

333

3

x

x

x

Removing the

24

251

51

51

831

22

x

x

x

x

x

,

82

835

8325

83124

:

false

Check

2,5

250

1070

961

31

31

2

2

22

x

xx

xx

xxx

xx

xx

11

3212

22

3515

:

check

Square root property (one x)

Completing the Square (two x’s)

Documents

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