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SECTION 3.1
Phrases
The difference of 4 and 3
( 4 - 3)
The sum of 4 and 3
( 4 + 3)
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The product of 4 and 3
( 4 · 3)
The quotient of 4 and 3
( 4  3)
a variable x, n, t, etc……
A number
Example: four times the sum of five and a number.
4
·
(
5
+
x )
4 (5 + x)
TWO SPECIAL CASES
A number subtracted from 3
X
-
7 less than a number
3
7
3- X
-
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.
Expressing one object in terms of another using a variable.
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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.
SECTION 3.2
Formulas:
D  rt
C
5
F  32
9
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?
Evaluating an Expression
1) Replace variables with parenthesis.
2) insert given numbers in parenthesis.
3) order of operations.
4
x  5 for x=3
3
1) 3x  4 for x=-2
2)
4) .75a 2  2.5a  2 for a=0
5) 3n 2  n  2 for n=2
3) 2 x 2  4 for x=-1
6)  4r 2  3r  1 for r=-2
SECTION 3.3
Expressions YouTube Video
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 and make them look pretty.
Multiplying
Multiply the numbers and write the letters
32 x 
 3x 2 y 
5r   2 3 ys 
2 x3 y 
4s3
5r  2 3rs
3x  2 y   4s  3
4s  3  3x  2 y 
DISTRIBUTING
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3x  2
52 x  1
  x  2
 53x  2
5r  6 2
5 − (3𝑥 − 2)
7 − 2(−5𝑥 + 6)
6𝑥 + (2𝑥 − 5)
5𝑥(3𝑟 − 2)
SECTION 3.4
Expressions YouTube Video
A)
5
C) 3
+
+ 2
3
B) 4 X + 7 X
+ 2
D) 8x - 5x² + 6x²
E) 3x  2 x  y
F)
G)
12a  A  a  8 A  2a
2)
9 AB  3 AB
4)
2
4
Y Y
3
9
1)
1
x
5
+
2
5
x
–m –n -8m +n
3) 3X-8Y-10X+4
5)
3 2 5 2
x  x
8
12
6) 0.25 x  3.2 y  9 x
Expression:
3x  2  4 5x  1
It’s to long, cut it down to size.
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1) Distribute
2) Combine like terms
Try:
 3  4 x  2  2  3x
3t  t  8
4  t  8
SECTION 1.6, 1.7
Linear Equations
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Question: Is 1 a solution to
3x  4  7 ? ___________why?_____________________________________
SIMPLE EQUATIONS:
X 3  2
2 X  4
 3 X  15
X
 3
2
Try:
X 7  2
x 5  4
8 X  24
2
X
7
SECTION 2.6
Linear Equations
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BASIC EQUATIONS:
1) Isolate the x
(x’s on one side, non x’s on the other)
(add and subtract)
3x  6  18
2) Divide
(Divide by the number next to the variable)
a) 𝟑𝒙 + 𝟓 = 𝟖
b) −𝟒 = 𝟓 − 𝟑𝒙
 x 5  7
SECTION 3.5
Linear Equations
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SIMPLIFYING A BASIC EQUATION:
1) Simplify
(simplify each side)
i) Distribute/Multiply
ii) Combine like term
 2  3x  8  18(1)
5 x  2   2 x  18  3
2) Isolate the x
3) Divide
Try:
a) −𝟐(𝟑𝒙 + 𝟓) = 𝟖
b) −𝟒(−𝟐) = 𝟑 − (−𝟓) − 𝟑𝒙
Linear Equations
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EQUATIONS WITH TWO OR MORE VARIABLES (Converting to a Basic equation)
1) Simplify
(simplify each side)
i) Distribute/Multiply
ii) Combine like term
3x  8  2 x  12
5( x  6)  2 x
2) Isolate the x
(pick an x side)
3) Divide
Try:
a) −𝟐𝒙 + 𝟓𝟎 = 𝟖𝒙 − 𝟏𝟎
b) −𝟒𝒙 = 𝟑(𝒙 + 𝟕)
SECTION 4.8
Linear Equations
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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
1
3
5
3
2
6
2
5
𝑥+ =
𝑥−
7
15
𝑥
11
3
5
= +
Try:
a)
3x
 5  2  x
5
SECTION 3.6 , 1.6, 1.7, 2.6
b)
3
x 1
x2 
5
3 15
c)
x
37 x
4
Phrases - Equations
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The following four phrases show operations between two things. The create quantities, parenthesis:
The difference of 4 and 3
( 4 - 3)
The sum of 4 and 3
( 4 + 3)
The product of 4 and 3
( 4 · 3)
The quotient of 4 and 3
( 4  3)
The product of 4 plus 3 and 7
4
 3  7
four times the sum of five and six.
4
·
(
5 + 6 )
4 (5 + 6)
TWO SPECIAL CASES
A number subtracted from 3
X
-
3
3- X
7 less than a number
7
-
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.
For the following.
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1) List the quantities and define a variable, 2) form a statement and equation, 3) solve and answer the
question.
1. In 1985, 595.4 billion cigarettes were smoked. This is 202.3 billion more cigarettes than were smoked in
2005. Find the number of cigarettes smoked in 2005.
1) List the quantities and define a variable.
2) Form a statement.
3) solve, answer, and say what this means.
2. In 2005, advertisers spent $6.3 billion on outdoor advertising. This is $3.7 billion more than advertisers
spent in 1990. Find the amount that advertisers spent on outdoor advertising in 1990.
1) List the quantities and define a variable.
2) Form a statement.
3) solve, answer, and say what this means.
3. According to the Census Bureau , per capita state taxes collected in a recent year averaged $2190. This
represents two and one-half times the average per capita income tax collected that year. Find the average per
capita income tax collected that year.
1) List the quantities and define a variable.
2) Form a statement.
3) solve, answer, and say what this means.
4. According to the Environmental Protection Agency, 58 million tons of waste was collected for recycling
in 2005. This is 2 million tons less than twice the amount of waste collected for recycling in 1990. Find the
amount of waste collected for recycling in 1990.
1) List the quantities and define a variable.
2) Form a statement.
3) solve, answer, and say what this means.
5. The total cost to paint the inside of a house was $2692. This cost included $250 for materials and $66 per
hour for labor. how many hours of labor were required?
1) List the quantities and define a variable.
2) Form a statement.
3) solve, answer, and say what this means.
6. Seven thousand dollars is divided into two scholarships. Twice the amount of the first scholarship is
$1000 less than the amount of the larger scholarship. What is the amount of the larger scholarship?
1) List the quantities and define a variable.
2) Form a statement.
3) solve, answer, and say what this means.
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.
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.
Word Problems-Consecutive Integers
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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.
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.