+
Midterm Review
+
+
Algebra Vocabulary

Algebraic Expression- a math sentence involving a variable
or unknown

Variable- a symbol for a letter we don’t know yet

Constant- a number on its own

Coefficient- a number used to multiply a variable

Operator- a symbol that shows an operation

Like Terms- are terms whose variables (and their
exponents) are the same. In other words, terms that are "like"
each other. (Note: the coefficients can be different)

Polynomial- an equation that has constants, variables and
the exponents; NEVER DIVISION
+ Rules
1.
You can only combine like terms
2.
Signs (+/-) are attached to the coefficient and variable
3.
- (-) = +
4.
Positive + Positive = Positive
5.
Positive + Negative = the sign of the larger digit
6.
Negative – Negative = the sign of the larger number
7.
Negative – Positive = negative
8.
Positive x Positive = Positive
9.
Positive x Negative = Negative
10.
Negative x Negative = Positive
11.
Positive
Positive = Positive
12.
Negative
Negative = Positive
13.
Negative
Positive = Negative
14.
Positive
Negative = Negative
+ Operation Key Words

Addition: sum, total, plus, in all, all together, and, increased
by

Subtraction: difference, less than, fewer, from, take away, are
not, remain, how many more, exceed, remain

Multiplication: times, each, in all, twice, product, factor,
multiple, multiplied by

Division: same, split, equal groups, separate, shared equally,
distribute, cut up
+
Write an expression

Identify the variable

Look for key words

Identify the operations
Example:
Take away 6 from 5 times of g
Subtract and Multiply
Answer: 5g-6
+
Write an expression



5 more than quotient of x and 3

Addition and Division

5 + x/3
9 less than 8 times of y

Subtraction and Multiplication

8y-9
Add 2 times of k and one-third

Addition and Multiplication

2 + 1k/3
+ Writing and Expression

Product of 5 and y

Quotient of 6 and a

Take away 2 from q

4 is added to p

3 times of x

One-half of q is added to 7

Two-third of the sum of 5 and m

Six-fifth of h is subtracted from 3

5 less than quotient of w and 7

9 minus h
+
Adding Polynomials

Line up like terms (this may involve re-arranging)
2x2 + 3xy+4
3x2+5xy+2

Add down the columns (Add like terms)
2x2 + 3xy+4
3x2+ 5xy +2
Note: the only aspect that changes is the coefficient

Answer: 5x2+ 8xy + 6
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Adding Polynomials
+
Adding Polynomials
+
Subtracting Polynomials

Line up like terms (this may involve re-arranging)
2x2 + 3xy+4
3x2+5xy+2

Multiply all terms in the second polynomial by -1
3x2+5xy+2

-3x2- 5xy -2
Add down the columns (Add like terms)
2x2 + 3xy +4
-3x2- 5xy -2
Note: the only aspect that changes is the coefficient

Answer: -1x2-2xy + 2
+
Subtracting Polynomials
+
Subtracting Polynomials
+
Distributive Property

The distributive property lets you multiply a sum by
multiplying each addend separately and then add the
products.
+
Distributive Property

5 (3 +2)

2( 13 +7)

10 (5 +5)

25 (4+6)

50 (2+5)
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Multiplying Exponents

When multiplying variables with exponents, the coefficients
are multiplied but the exponents are added together
Example:
(3x3) (10x4)
(3 x 10) ( x3+4)
30x7

However, we can NOT simplify (x4)(y3), because the bases are
different
+
Multiplying Exponents
+
Dividing Exponents

Dividing exponents with same base
For exponents with the same base, we subtract the exponents
and leave the bases the same:

a n / a m = a n-m
Example:

26 / 23 = 26-3 = 23 = 2·2·2 = 8
+
Dividing Exponents
+
Dividing Exponents

Dividing variables with exponents
For exponents with the same base, we can subtract the
exponents:

xn / xm = xn-m

Example:

x5 / x3 = (x·x·x·x·x) / (x·x·x) = x5-3 = x2
+
Dividing Exponents with
Coefficients and Variables
Divide the coefficients
Remember: top exponent minus bottom exponent.
Remember: negative exponents can be written as a
fraction.
+
Dividing Exponents with
Coefficients and Variables
The exponents are subtracted for the bases that are the SAME.
The numbers in front, the coefficients, are divided.
+
Dividing Exponents with
Coefficients and Variables

Notice what happened to the bases with the same
exponents – a0 which is equivalent to 1.
+
Distributive Property and Algebra

Still distributing the number that is outside of the parenthesis
to what is inside, except we can only combine like terms.
5x+10
15x2+30x
15x2+10x+30
+
Distributive Property and Algebra
+
FOIL Method

FOIL stands for:

First - Multiply the first term in each set of parentheses

Outer - Multiply the outer term in each set of parentheses

Inner - Multiply the inner term in each set of parentheses

Last - Multiply the last term in each set of parentheses
+
FOIL

Simplify (3+7x) (6+2x)

FIRST

We'll start by multiplying the first term in each set of
parentheses and then marking down the answer below the
problem.
+
FOIL

Simplify (3+7x) (6+2x)

Outside

Now we will multiply the outer terms and again mark down
the answer below the problem.
+
FOIL

Simplify (3+7x) (6+2x)

Inside

Now we will multiply the inner terms and again mark down
the answer below the problem.
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FOIL

Simplify (3+7x) (6+2x)

Last

Now we will multiply the last terms and again mark down the
answer below the problem.
+
FOIL

Simplify (3+7x) (6+2x)

Combine Like Terms

We see that 6x + 42x = 48x, therefore our answer is:
+
FOIL Practice
+
What if we can’t use FOIL

The FOIL Method cannot always be used to multiply two sets
of parentheses. This is the case with the problem below.
The second set of parentheses has 3
terms instead of two. For this, we will use
another method.
+
Multiplying Polynomials

First, split the problem by multiplying each term in the left
polynomial by the entire second polynomial as shown below.
+
Multiplying Polynomials

Then use the distributive property to simplify each.
+
Multiplying Polynomials

Finally, combine like terms.
+
Multiplying Polynomials
1. Separate and rewrite
4a (6a2-a+2) + 2(6a2-a+2)
2. Use the Distributive Property
(24a3-4a2+8a)+ (12a2-2a+4)
3. Combine Like Terms
24a3-8a2+6a+4
+
Multiplying Polynomials
+
You Try
+
Addition
X + 4 = 10

Get all variables on one side

Get all constants on one side

Find X
To do this you subtract the constants
X + 4 = 10
-4
-4
X+0=6
X= 6
+
Practice
X+ 29 = 50
25 + d = 13
36+y = 45
15 + v = 2
+
Subtraction
X - 4 = 10

Get all variables on one side

Get all constants on one side

Find X
To do this you add the constants
X - 4 = 10
+4
+4
X + 0 = 14
X= 14
+
Practice
X- 48 = 150
25 - d = 13
96-y = 45
v - 25 = 200
+
Multiplication
4x = 20

Get all variables on one side

Get all constants on one side

Find X
To do this you divide by the coefficient
4x = 20
4
4
x= 5
+
Practice
3X = 150
-25d = 100
6y = 48
-5v = 200
-5x=-75
+
Division
x/4 =20

Get all variables on one side

Get all constants on one side

Find X
To do this you multiply by the coefficient you are dividing by
x/4 = 20
(4) x/4 = 20 (4)
4x/4= 80
X=80
+
Division
100/x =20

Get all variables on one side

Get all constants on one side

Find X
To do this you multiply by the variable you are dividing by and repeat the steps
for multiplication
100/x = 20
(x) 100/x = 20 (x)
100x/x= 20
100=20x
20 20
X=5
+
Practice
x/33 = 11
-25/d = 5
y/6 = 48
500/v = 20
+
Factoring Polynomials

Reverse Foil

Analyze each term in an expression and determine what was
multiplied together
• What multiplied together has a product of x2?
(x) (x)
• What multiplied together gives you a negative 15 but when
added together results in positive 2?
(5) (-3)
= (x+5) (x-3)
+
Factoring Polynomials
+
Addition
X + 4 < 10

Get all variables on one side

Get all constants on one side

Find X
To do this you subtract the constants
X + 4 < 10
-4
-4
X+0<6
X< 6
Rule applies if it is greater than, greater than or equal too, or less than and
equal too
+
Practice
X+ 29 < 50
25 + d < 13
36+y > 45
15 + v > 2
+
Subtraction
X – 4 < 10

Get all variables on one side

Get all constants on one side

Find X
To do this you add the constants
X – 4 < 10
+4
+4
X + 0 < 14
X< 14
+
Subtraction

If the variable is negative, you must divide by negative one
and the sign of the inequality is changed
3 - x <5
-3
-3
-x < 2
-1
-1
X>-2
Rule applies if it is greater than, greater than or equal too, or
less than and equal too
+
Practice
X- 48 > 150
25 – d < 13
96-y < 45
v – 25 > 200
+
Multiplication
4x < 20

Get all variables on one side

Get all constants on one side

Find X
To do this you divide by the coefficient
4x < 20
4
4
x< 5
Rule applies if it is greater than, greater than or equal too, or less
than and equal too
+
Multiplication
-4x < 20

Get all variables on one side

Get all constants on one side

Since it is a negative coefficient we flip the sign

Find X
To do this you divide by the coefficient
-4x < 20
-4
- 4
x> - 5
Rule applies if it is greater than, greater than or equal too, or less
than and equal too
+
Practice
3X > 150
-25d < 100
6y < 48
-5v > 200
+
Division
x/4 <20

Get all variables on one side

Get all constants on one side

Find X
To do this you multiply by the coefficient you are dividing by
x/4 < 20
(4) x/4 < 20 (4)
4x/4< 80
X<80
+
Division
-3x/4 <20

Get all variables on one side

Get all constants on one side

Find X

If the variable is attached to a negative coefficient you must switch the sign
To do this you multiply by the coefficient you are dividing by
-3x/4 < 20
(4/3)-3 x/4 < 20 (4/3)
-12x/-12 < 80/-12
X> - 80/12
Rule applies if it is greater than, greater than or equal too, or less than and equal
too
+
Division
100/x <20

Get all variables on one side

Get all constants on one side

Find X
To do this you multiply by the variable you are dividing by and repeat the steps for
multiplication
100/x < 20
(x) 100/x < 20 (x)
100x/x< 20
100<20x
20
20
X<5
Rule applies if it is greater than, greater than or equal too, or less than and equal too
+
Practice
x/33 < 11
-5d/3 < 15
-y/6 > 48
500/v > 20
+
When solving inequalities we
graph the solution

If the sign is less than or greater than:

We draw a umber line

Put a circle above the solved value

Then draw a line from that circle to indicate the direction of
all possible answers
+
When solving inequalities we
graph the solution

If the sign is less than or equal too or greater than or equal
too:

We draw a umber line

Put a circle above the solved value

Then draw a line from that circle to indicate the direction of
all possible answers
+
Graph the following
+
Solve and Graph
5-X >10
-2X >12
+
+
1. Get all numbers
and variables on
different sides
2. Get the variable
alone
3x+5=12
-5 -5
3x=7
3 3
X= 7/3
+
1. Get all numbers
and variables on
different sides
2. Get the variable
alone
4y-8=12
+8 +8
4y=20
4 4
X= 5
+
1. Get all numbers
and variables on
different sides
2. Get the variable
alone
5m-4=-25
+4 +4
5m=-21
5
5
X= -21/5
+
Practice
+
1. Combine Like
Terms
2. Get all numbers
and variables on
different sides
3. Get the variable
alone
7x-3x-8=24
4x-8=24
+8 +8
4x=32
4 4
X= 8
+
1. Combine Like
Terms
2. Get all numbers
and variables on
different sides
3. Get the variable
alone
2/5x-1/5x+9=-1
1/5x+9=-1
-9 -9
(5/1)1/5x=-10 (5/1)
X= -50
+
1. Combine Like
Terms
2. Get all numbers
and variables on
different sides
3. Get the variable
alone
25x-16x-24=-65
9x-24=-65
+24 +24
9x = -41
9
9
X= -41/9
+
Practice
+
1. Use the
Distributive
Property
2. Combine Like
Terms
3. Get all numbers
and variables on
different sides
4. Get the variable
alone
5x+3(x+4)=28
5x+3x+12=28
8x+12=28
-12 -12
8x = 16
8
8
X= 2
+
1. Use the
Distributive
Property
2. Combine Like
Terms
3. Get all numbers
and variables on
different sides
4. Get the variable
alone
4x+3(x-2)=21
4x+3x-6=21
7x-6=21
-6 -6
7x = 15
7
7
X= 15/7
+
1. Use the
Distributive
Property
2. Combine Like
Terms
3. Get all numbers
and variables on
different sides
4. Get the variable
alone
2x+5(x-9)=27
2x+5x-45=27
7x-45=27
-45 -45
7x = -28
7
7
X= -4
+
Practice
+
Multi-Step Equations

Just as with solving one-step or two-step or any equation, one
goal in solving an equation is to have only variables on one
side of the equal sign and numbers on the other side of the
equal sign. The other goal is to have the number in front of
the variable equal to one.

The strategy for getting the variable by itself with a
coefficient of 1 involves using opposite operations.

The most important thing to remember in solving a linear
equation is that whatever you do to one side of the equation,
you MUST do to the other side.
+
Multi-Step Equations
Solve
+
Multi-Step Equations
We must first put the variables on the same side.
Let’s move the 2x from the right side to the left side
by subtracting 2x from both sides.
+
Multi-Step Equations
Now we bring the constants over to the same side.
Since we have a positive 9, we subtract 9 from both
sides.
+
Multi-Step Equations
Now to get the variable to have a coefficient of 1, we
need to divide both sides of our coefficient by 2.
+
Multi-Step Equation
+
Multi-Step Equation
1. Move the Variables
2. Move the Constants
3. Get the Variable to have
an exponent of 1
+
Multi-Step Equations
1. Move the Variables
2. Move the Constants
3. Get the Variable to have
an exponent of 1
+
Multi-Step Equations
1. Distributive Property
2. Combine like terms
3. Move the Variables
4. Move the Constants
5. Get the Variable to have
an exponent of 1
+
Multi-Step Equations
1. Distributive Property
2. Combine like terms
3. Move the Variables
4. Move the Constants
5. Get the Variable to have
an exponent of 1
+
Multi-Step Equations
+
2x
1. Get all numbers
and variables on
different sides
2. Get the variable
alone
-4 -4
2x
20
2 2
X = 10
+
1. Get all numbers
and variables on
different sides
2. Get the variable
alone
5x 10
25
+10 +10
5x
35
5 5
X= 7
+
b
1. Get all numbers
and variables on
different sides
2. Get the variable
alone
-b
-1
b
+2 +2
10
-1
10
+
Practice
+
1. Combine Like
Terms
2. Get all numbers
and variables on
different sides
3. Get the variable
alone
n
n+6
3 -3n + 6
-6
-6
-3 -3n
-3 -3
n -6
+
x
1. Combine Like
Terms
2. Get all numbers
and variables on
different sides
3. Get the variable
alone
x
12x + 2
12x
12
x 1
-2 -2
2
12
+
Practice
+
1. Use the
Distributive
Property
2. Combine Like
Terms
3. Get all numbers
and variables on
different sides
4. Get the variable
alone
(p
) - 3p +
3
-3p-3-3p+3
-6p
-6 -6
p 3
+
1. Use the
Distributive
Property
2(x
2. Combine Like
Terms
3. Get all numbers
and variables on
different sides
4. Get the variable
alone
) + 2x + 3 37
2x+2+2x+3 37
4x+5 37
-5 -5
4x 32
4 4
X 8
+
Practice
+
Absolute Value

Solve the equation twice

First time as if the absolute value symbols were parentheses

Second time by changing all the sides inside the absolute
value sign
+
Absolute Value
+
Absolute Value
+
Proportions

Cross multiply and follow the steps for a multi-step equation
+ Proportions