1

Section 1.1 Linear Equations

YouTube Video

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

SIMPLE EQUATIONS:

23 X 42 X 153 X 32

X

Try:

27 X 45 x 248 X 7

2

X

2

Linear Equations

YouTube Video

BASIC EQUATIONS:

1) Isolate the x 1863 x 75 x

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

(add and subtract)

2) Divide

(Divide by the number next to the variable)

a) b)

3

318225 xx

Linear Equations YouTube Video

SIMPLIFYING A BASIC EQUATION:

1) Simplify )1(18832 x

(simplify each side)

i) Distribute/Multiply

ii) Combine like term

2) Isolate the x

3) Divide

Try:

a) b)

4

Linear Equations

YouTube Video EQUATIONS WITH TWO OR MORE VARIABLES (Converting to a Basic equation)

1) Simplify 12283 xx xx 2)6(5

(simplify each side)

i) Distribute/Multiply

ii) Combine like term

2) Isolate the x

(pick an x side)

3) Divide

Try:

a) b)

5

Linear Equations

YouTube Video EQUATIONS INVOLVING FRACTIONS:

1) Simplify (simplify each side)

i) Distribute/Multiply

ii) Combine like term

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

2) Isolate the x

(pick an x side)

3) Divide

Try:

a) xx

255

3 b)

15

1

32

5

3

xx c) x

x 73

4

6

Section 1.2

Formulas - Solving for a Given Variable

YouTube Video

Simplify Isolate Divide Fraction--- add/subtract both sides

Parenthesis---

Like terms---

Solving for a given variable (formulas)

a) DPS , for D b) bBhA 2

1, for b c)

2d

kL , for k d) yarr 32 , for r

tPA Pr , for P 329

5 FC , for F CRP , for C|

7

rtD 329

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

You Tube Video

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 -13 ?

3) For the month of June, a florist’s cost of doing business was $3,795. If June’s revenues totaled $5,115,

what was her profit for the month of June?

4) You purchase a shoe for at a cost of $10 at a yard sale and you sell the shoe for a loss of $2. What was

the selling price or revenue?

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Section 1.3 Phrases YouTube Video

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)

A number a variable x, n, t, etc……

Example: four times the sum of five and a number.

4 · ( 5 + x ) 4 (5 + x)

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 minus 2 and 5

7. 4 minus the sum of 5 and 2. 8. The difference of 6 and 4 9. 7 less than a number

10. 7 less a number. 11. twice the price p. 12. half of a number

13. 7 divided by a number 14. Exceeds x by 7 15. 5 increased by t

16. 6 minus x 17. The product of 3 and the sum of a number and 4.

18. The sum of 5 and the square of t. 19. one third of the square of x. 20. 4 less than the cube of y.

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Phrases - Equations

YouTube Video

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 7 734

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 a number is 4 more than the number

2. The sum of 7 and 8 is the number of dogs in a car.

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.

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Expressing one object in terms of another using a variable. YouTube Video

1. If the length is 4 ft more than the width. Express the length and width using one variable.

Length=____________________ width=____________________________

2. There are 4 times as many apples as pears in a bowl. Express the number of apples and pears using one

variable.

Number of apples=____________________ Number of pears=____________________________

3. The base of a triangle is four less than the height. Express the height and base using one variable

=____________________ =____________________________

4. One cyclist rides six miles per hour faster than another cyclist. Express the speed of the faster cyclist in

terms of the speed of the slower cyclist.

5. The planet Saturn as 7 more moons than Jupiter. Express the number of moons Saturn has in terms of the

number of moons Jupiter has.

6. The sale price of a suit is 3/4 ths of the original price. Express the sale price of the suit in terms of the

original price.

7. A bank contains 12 coins made up of dimes and nickels. Express the number of nickels in terms of the

number of dimes.

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Word Problems-Translations-Two or more quantities YouTube Video

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 variable. (list your quantities or underline them)

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

b) Form an equation. (write a statement or draw a picture)

c) Solve the equation.

d) Review your answers.

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

the garden.

a) Define your variable. (list your quantities or underline them)

Width=__________________ Length=__________________

b) Form an equation. (write a statement or draw a picture)

c) Solve the equation.

c) Solve the equation.

d) Review your answers.

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3) A box containing pears, apples, and oranges weighs 73 pounds ignoring the box. If the weight of pears is

the same as the weight of apples and the weight of oranges is 4 lbs more than the weight of apples, then find

the weight of each type of fruit.

a) Define your variable. (list your quantities or underline them)

Weight of pears:__________, Weight of apples:__________, Weight of oranges:__________

b) Form an equation. (write a statement or draw a picture)

c) Solve the equation.

d) Review your answers.

*4) A real estate agent sold two homes and received commissions totaling $6000. The agent’s commission

on one home was one-third of the commission on the second home. Find the agents commission on each

home.

a) Define your variable. (list your quantities or underline them)

Commission on one home=_______________, Commission on second home=_______________.

b) Form an equation. (write a statement or draw a picture)

c) Solve the equation.

d) Review your answers.

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Simple Interest= (simple interest rate)%TotalYears

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

Interest rate Principle Years Simple Interest

=

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

Interest rate Principle Years Simple Interest

=

3. a) Jimbo invests x in an account making 3%. What is the expression that represents his interest after 1

year?

Interest rate Principle Years Simple Interest

=

b) If he invests x dollars in a 4% account, then what is the expression that represents his interest after 1

year?

Interest rate Principle Years Simple Interest

=

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

3. a) Sarah Invest $3000 into an account for 1 year at 4%. She also invests $5000 in an account for 1 year at

8%. What is her total amount invested and her total interest for the year?

Interest rate Principle Years Simple Interest

14

Word Problems-Simple Interest YouTube Video

1) 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?

a) Define your variable. Principle Rate = Interest

1st account

2nd account

Total

b) Form an equation and c) Solve the equation.

d) Review your answers.

2) 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?

a) Define your variable. Principle Rate = Interest

1st account

2nd account

Total

b) Form an equation and c) Solve the equation.

d) Review your answers.

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3) Thomas invested $4000 into two accounts. One account earning a 4% interest rate and the other account

earning a 3% interest rate. If the total interest for the year was $145, then how much was invested in each

account?

a) Define your variable. Principle Rate = Interest

1st account

2nd account

Total

b) Form an equation and c) Solve the equation.

d) Review your answers.

4) Julia invested some money in and account earning 7%. She then invested 4 times as much money in a

10% account as she did in the 7% account. If her total interest for the year was $940, then how much was

invested at each rate?

a) Define your variable. Principle Rate = Interest

1st account

2nd account

Total

b) Form an equation and c) Solve the equation.

d) Review your answers.

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Mixture Problems

YouTube Video

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%?

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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?

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Section 1.4

Coin/Stamps Problems

YouTube Video

25 Quarters= cents

1) Sylvia 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

2) 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

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3) Lucinda found some stamps while looking under her sofa pillows. She found three times as many 5 cent

stamps as 32 cent stamps and five more 15 cent stamps as 32 cent stamps. If the total value of the stamps

was $3.85, then how many of each stamp did she find?

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

4) Thomas has a bank that contains pennies and dimes. If the total number of coins is 60 with a total value

of $3.03, then how many of each type of coin is in the bank.

5) 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.

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Uniform Motion problems warm-up YouTube Video (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 in terms of t?

Rate Time Distance

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

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 in terms of x?

Rate Time Distance

How far will Leann travel in 3 hours in terms of x?

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 in terms of t?

Rate Time Distance

How far has Silvia traveled in terms of t?

Rate Time Distance

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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) YouTube Video

1) It’s 12pm. Super Snail and Sergeant Salt are 12 inches apart and start traveling towards each other. If

Super Snail travels at 4 inches per minute and Sergeant Salt travels at 2 inches per minute, then when will

they meet?

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?

rtd train 1 rtd train 2

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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.

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) 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:

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Word Problems-Consecutive Integers

YouTube Video 1) Three consecutive positive integers have a sum of 36. Find the integers.

a) Define your variables.

Integer 1is _________________, Integer 2 is__________________, Integer 3 is____________________

b) Form and equation.

_______+________+___________=36

c) Solve the equation.

d) Review your answers.

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

a) Define your variables.

Integer 1is _________________, Integer 2 is__________________, Integer 3 is____________________

b) Form and equation.

_______+________+___________=

c) Solve the equation.

d) Review your answers.

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

integer.

a) Define your variables.

Integer 1is _________________ Integer 2 is__________________

b) Form and equation.

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

c) Solve the equation.

d) Review your answers.

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

integers.

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Section 1.5

Linear Inequalities YouTube Videos

<,> ,

4x x>-5 x<0

Graph

Interval notation

1)

2x

Interval notation:

2)

6x

Interval notation:

3)

62 x

Interval notation:

4)

175 x

Interval notation:

26

Solving linear inequalities YouTube Video

If you divide or multiply by a negative then

Solve and graph.

413 x 322

313

3

2 kk )4(2)4(35 xxx

16426 z

Answer:

Interval Notation:

105

122

x

Answer:

Interval Notation:

27

Inequalities YouTube Videos

< >

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,

List some numbers between 3 and 10 x is between 3 and 10

28

Word Problems Involving Inequalities YouTube Video

1. 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?

2. 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 greater than that the $250

flat fee?

3. Suppose that you have ordered one medium onion ring and one 16-ounce chocolate shake from Burger

King. The onion rings have 16 grams of fat and the shake has 8 grams of fat. Each cheeseburger has 21

grams of fat. How many cheeseburgers can you order and still keep the total fat content of the meal to at

most 87 grams?

4. Two times the difference between a number and eight is less than or equal to five times the sum of the

number and four. Find the smallest number that will satisfy the inequality.

5. George earns $1000 per month plus 5% commission on the amount of sales. George needs to earn a

minimum of $3,200 per month. What amount of sales does George need so that his monthly income is at

least $3,200?

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Section 1.6

Inequalities with

/

YouTube Videos

2x

6x

62 xorx

62 xandx

Answer

Interval Notation

Answer

Interval Notation

Answer

Interval Notation

30

Section 1.7

Solving Equations Involving Absolute Values

YouTube Video Absolute Value- Distance from Zero

4x 2x

Solve.

74 x 1532 x

753 x 2575 x

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Solving Inequalities Involving Absolute Values

YouTube Video Absolute Value- Distance from Zero

2x 4x

Solve.

, Greator , lessthand

74 x 74 x

1523 x 1523 x

32

Solving Equations and Inequalites Involving Absolute Values

(Special Cases)

YouTube Video

Look at the following and think about the answer. If there is no solution, then say no

solution. If there are infinitely many solutions, then use the proper interval notation.

452 x 415 x 415 x

072 x 0147 x 02 x 053 x