Transcript (3)(a) - Reeths-Puffer Schools

Expressions & Equations
2013-01-23
www.njctl.org
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Table of Contents
Commutative and Associative Properties
Combining Like Terms
Click on a topic to go to
that section.
The Distributive Property and Factoring
Simplifying Algebraic Expressions
Inverse Operations
One Step Equations
Two Step Equations
Multi-Step Equations
Distributing Fractions in Equations
Translating Between Words and Equations
Using Numerical and Algebraic Expressions and Equations
Graphing & Writing Inequalities with One Variable
Simple Inequalities involving Addition & Subtraction
Simple Inequalities involving Multiplication & Division
Common Core Standards: 7.EE.1, 7.EE.3, 7.EE.4
Commutative and
Associative Properties
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Commutative Property of Addition: The order in which
the terms of a sum are added does not change the sum.
a+b=b+a
5+7=7+5
12= 12
Commutative Property of Multiplication: The order in
which the terms of a product are multiplied does not
change the product.
ab = ba
4(5) = 5(4)
Associative Property of Addition: The order in which the
terms of a sum are grouped does not change the sum.
(a + b) + c = a + (b + c)
(2 + 3) + 4 = 2 + (3 + 4)
5+4=2+7
9=9
The Commutative Property is particularly useful when
you are combining integers.
Example:
-15 + 9 + (-4)=
-15 + (-4) + 9=
-19 + 9 =
-10
Changing it this way allows for the
negatives to be added together first.
Associative Property of Multiplication: The order in
which the terms of a product are grouped does not
change the product.
1
Identify the property of -5 + 3 = 3 + (-5)
A
B
C
D
Commutative Property of Addition
Commutative Property of Multiplication
Associative Property of Addition
Associative Property of Multiplication
2
Identify the property of a + (b + c) = (a + c) + b
A
B
C
D
Commutative Property of Addition
Commutative Property of Multiplication
Associative Property of Addition
Associative Property of Multiplication
3 Identify the property of (3 * (-4)) * 8 = 3 * ((-4) * 8)
A
B
C
D
Commutative Property of Addition
Commutative Property of Multiplication
Associative Property of Addition
Asociative Property of Multiplication
Discuss why using the Commutative Property would
be useful with the following problems:
1. 4 + 3 + (-4)
2. -9 x 3 x 0
3. -5 x 7 x -2
4. -8 + 1 + (-6)
Combining Like Terms
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An Algebraic Expression - contains numbers, variables
and at least one operation.
Like terms: terms in an expression that have the
same variable raised to the same power
Examples:
LIKE TERMS
NOT LIKE TERMS
2
6x and 2x
6x and 2x
5y and 8y
5x and 8y
2
2
4x and 7x
2
2
4x y and 7xy
4
Identify all of the terms like 2x
A
B
C
D
E
5x
2
3x
5y
12y
2
5
Identify all of the terms like 8y
A
B
C
D
E
9y
2
4y
7y
8
-18x
6
Identify all of the terms like 8xy
A
B
C
D
E
8x
2
3x y
39xy
4y
-8xy
7
Identify all of the terms like 2y
A
B
C
D
E
51w
2x
3y
2w
-10y
8
Identify all of the terms like 14x
A
B
C
D
E
-5x
2
8x
2
13y
x
2
-x
2
If two or more like terms are being added or
subtracted, they can be combined.
To combine like terms add/subtract the coefficient
but leave the variable alone.
7x +8x =15x
9v-2v = 7v
Sometimes there are constant terms that can be
combined.
9 + 2f + 6 =
9 + 2f + 6 =
2f + 15
Sometimes there will be both coeffients and constants to
be combined.
3g + 7 + 8g - 2
11g + 5
Notice that the sign before a given term goes with the
number.
Try These:
1.) 2b +6g(3) + 4f + 9f
2.) 9j + 3 + 24h + 6 + 7h + 3
3.) 7a + 4 + 2a -1 9 + 8c -12 + 5c
4.) 8x + 56xy + 5y
9
8x + 3x = 11x
A
B
True
False
10 7x + 7y = 14xy
A
B
True
False
11
2x + 3x = 5x
A
B
True
False
12
9x + 5y = 14xy
A
B
True
False
13
6x + 2x = 8x
A
B
2
True
False
14
-15y + 7y = -8y
A
B
True
False
15
-6 + y + 8 = 2y
A
B
True
False
16
-7y + 9y = 2y
A
B
True
False
17 9x + 4 + 2x =
A
B
C
D
15x
11x + 4
13x + 2x
9x + 6x
18 12x + 3x + 7 - 5
A
B
C
D
15x + 7 - 5
13x
17x
15x + 2
19 -4x - 6 + 2x - 14
A
B
C
D
-22x
-2x - 20
-6x +20
22x
The Distributive Property
and Factoring
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An Area Model
Imagine that you have two rooms next to
each other. Both are 4 feet long. One is 7
feet wide and the other is 3 feet wide .
How could you express
the area of those two
rooms together?
4
7
3
4
4
4
7+ 3
7
3
You could multiply 4 by 7,
then 4 by 3 and add them
You could add 7 + 3
and then multiply by 4
OR
4(7+3)=
4(10)=
40
4(7) + 4(3) =
28 + 12 =
40
2
Either way, the area is 40 feet :
An Area Model
Imagine that you have two rooms next to each
other. Both are 4 yards long. One is 3 yards
wide and you don't know how wide the other is.
How could you express
the area of those two
rooms together?
4
x
3
You cannot add x and 3
because they aren't like
terms, so you can only do it
by multiplying 4 by x and 4 by
3 and adding
4
x+ 3
4(x) + 4(3)=
4x + 12
The area of the two rooms is
4x + 12
(Note: 4x cannot be combined with 12)
The Distributive Property
Finding the area of the rectangles demonstrates the
distributive property. Use the distributive property when
expressions are
written like so: a(b + c)
4(x + 2)
4(x) + 4(2)
4x + 8
The 4 is distributed to each term of the sum (x + 2)
Write an expression equivalent to:
5(y + 4)
5(y) + 5(4)
5y + 20
Remember to distribute the 5 to the y and the 4
6(x + 2)
3(x + 4)
4(x - 5)
7(x - 1)
The Distributive Property is often used to eliminate the
parentheses in expressions like 4(x + 2). This makes it
possible to combine like terms in more complicated
expressions.
Be careful with
EXAMPLE:
your signs!
-2(x + 3) = -2(x) + -2(3) = -2x + -6 or -2x - 6
3(4x - 6) = 3(4x) - 3(6) = 12x - 18
-2 (x - 3) = -2(x) - (-2)(3) = -2x + 6
TRY THESE:
3(4x + 2) =
-1(6m + 4) =
-3(2x - 5) =
Keep in mind that when there is a negative sign on the
outside of the parenthesis it really is a -1.
For example:
-(2x + 7) = -1(2x + 7) = -1(2x) + -1(7) = -2x - 7
What do you notice about the original problem and its
answer?
Remove to see answer.
The numbers are turned to their opposites.
Try these:
-(9x + 3) =
-(-5x + 1) =
-(2x - 4) =
-(-x - 6) =
20
4(2 + 5) = 4(2) + 5
A
B
True
False
21
8(x + 9) = 8(x) + 8(9)
A
B
True
False
22
-4(x + 6) = -4 + 4(6)
A
B
True
False
23
3(x - 4) = 3(x) - 3(4)
A
B
True
False
24 Use the distributive property to rewrite the
expression without parentheses
3(x + 4)
A
B
C
D
3x + 4
3x + 12
x + 12
7x
25 Use the distributive property to rewrite the
expression without parentheses
5(x + 7)
A
B
C
D
x + 35
5x + 7
5x + 35
40x
26 Use the distributive property to rewrite the
expression without parentheses
(x + 5)2
A
B
C
D
2x + 5
2x + 10
x + 10
12x
27 Use the distributive property to rewrite the
expression without parentheses
3(x - 4)
A
B
C
D
3x - 4
x - 12
3x - 12
9x
28 Use the distributive property to rewrite the
expression without parentheses
2(w - 6)
A
B
C
D
2w - 6
w - 12
2w - 12
10w
29 Use the distributive property to rewrite the
expression without parentheses
-4(x - 9)
A
B
C
D
-4x - 36
x - 36
4x - 36
-4x + 36
30 Use the distributive property to rewrite the
expression without parentheses
5.2(x - 9.3)
A
B
C
D
-5.2x - 48.36
5.2x - 48.36
-5.2x + 48.36
-48.36x
31 Use the distributive property to rewrite the
expression without parentheses
A
B
C
D
We can also use the Distributive Property in reverse.
This is called Factoring.
When we factor an expression, we find all numbers or
variables that divide into all of the parts of an
expression.
Example:
7x + 35
Both the 7x and 35 are divisible by 7
7(x + 5)
By removing the 7 we have factored the
problem
We can check our work by using the distributive
property to see that the two expressions are equal.
We can factor with numbers, variables, or both.
2x + 4y = 2(x + 2y)
9b + 3 = 3(3b + 1)
-5j - 10k + 25m = -5(j + 2k - 5m) *Careful of your signs
4a + 6a + 8ab = 2a(2 + 3 + 4b)
Try these:
Factor the following expressions:
1.) 6b + 9c =
2.) -2h - 10j =
3.) 4a + 20ab + 12abc =
32 Factor the following: 4p + 24q
A
B
C
D
4 (p + 24q)
2 (2p + 12q)
4(p + 6q)
2 (2p + 24q)
33 Factor the following: 5g + 15h
A
B
3(g + 5h)
5(g + 3h)
C
D
5(g + 15h)
5g (1 + 3h)
34 Factor the following: 3r + 9rt + 15rx
A
B
C
D
3(r+ 3rt + 5rx)
3r(1 + 3t + 5x)
3r (3t + 5x)
3 (r + 9rt + 15rx)
35 Factor the following: 2v+7v+14v
A
B
C
D
7(2v + v + 2v)
7v(2 + 1 + 2)
7v (1 + 2)
v(2 + 7 + 14)
36 Factor the following: -6a - 15ab - 18abc
A
B
C
D
-3a(2 + 5b + 6bc)
3a(2+ 5b + 6bc)
-3(2a - 5b - 6bc)
-3a (2 -5b - 6bc)
-
What divides into the expression: -5n - 20mn - 10np
-
If a regular pentagon has a perimeter of 10x + 25,
what does each side equal?
Simplifying Algebraic
Expressions
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Now we will use what we know about combining like
terms and the distributive property to simplify
algebraic expressions.
Remember, like terms have the same variable and
same exponent.
To simplify:
4 + 5(x + 3)
First Distribute
4 + 5(x) + 5(3)
4 + 5x + 15
Then combine Like Terms
5x + 19
Notice that when combining like terms, you add/subtract
the coefficients but the variable remains the same.
Remember that you can combine coefficient or constant
terms.
37
7x +3(x - 4) = 10x - 4
A
B
True
False
38
8 +(x + 3)5 = 5x + 11
A
B
True
False
39
4 +(x - 3)6 = 6x -14
A
B
True
False
40
2x + 3y + 5x + 12 = 10xy + 12
A
B
True
False
41
2
2
2
5x + 2x + 7(x + 1) + x = 6x + 9x + 7
A
B
True
False
42
3
2
2
3
2
2x + 4x + 6(x + 3x) + x = 2x + 10x + 4x
A
B
True
False
43 The lengths of the sides of home plate in a
baseball field are represented by the
expressions in the accompanying figure.
yz
y
x
A
B
C
D
y
x
Which expression represents
the perimeter of the figure?
5xyz
2
3
x +y z
2x + 3yz
2x + 2y + yz
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17,
June, 2011.
44 A rectangle has a width of x and a length that
is double that. What is the perimeter of the
rectangle?
A
4x
B 6x
C 8x
D 10x
Inverse Operations
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What is an equation?
An equation is a mathematical statement containing an
equal sign to show that two expressions are equal.
2+3=5
9–2=7
5+3=1+7
An algebraic equation is just an equation that has
algebraic symbols in one or both of the expressions.
4x = 24
9 + h = 15
Equations can also be used to state the equality of two
expressions containing one or more variables.
In real numbers we can say, for example, that for any
given value of x it is true that
4x + 1 = 13
x = 3 because
4(3) + 1 = 13
12 + 1 = 13
13 = 13
An equation can be
compared to a
balanced scale.
Both sides need to
contain the same
quantity in order for it
to be "balanced".
For example, 9+ 11 = 6 + 14 represents an equation
because both sides simplify to 20.
9 + 11 = 6 + 14
20 = 20
Any of the numerical values in the equation can be
represented by a variable.
Examples:
15 + c = 25
x + 10 = 25
15 + 10 = y
When solving equations, the goal is to isolate the variable
on one side of the equation in order to determine its value
(the value that makes the equation true).
In order to solve an equation containing a variable,
you need to use inverse (opposite/undoing)
operations on both sides of the equation.
Let's review the inverses of each operation:
Addition
Multiplication
Square
Subtraction
Division
Square Root
There are two questions to ask when solving an
equation:
*What operation is in the equation?
*What is the inverse of that operation (This will be the
operation you use to solve the equation.)?
A good phrase to remember when doing equations is:
Whatever you do to one side of the equation, you do to
the other.
For example, if you add three on one side of the equal
sign you must add three to the other side as well…
to keep the equation in balance.
To solve for "x" in the following equation...
x + 7 = 32
Determine what operation is being shown (in this case, it
is addition). Do the inverse to both sides (in this case, it
is subtraction).
x + 7 = 32
- 7 -7
x = 25
In the original equation, replace x with 25 and see if it
makes the equation true.
x + 7 = 32
25 + 7 = 32
32 = 32
For each equation, write the inverse operation needed to
solve for the variable.
move
a.) y +7 = 14 subtract
7
b.) a - 21 = 10
c.) 5s = 25
d.) x = 5
12
divide
move by 5
addmove
21
multiply
by 12
move
Think about this...
To solve c - 3 = 12
Which method is better? Why?
Kendra
Ted
Added 3 to each side
of the equation
Subtracted 12 from each side,
then added 15.
c - 3 = 12
+3 +3
c = 15
c - 3 = 12
-12 -12
c - 15 = 0
+15 +15
c = 15
45 What is the inverse operation needed to solve
this equation?
2x = 14
A Addition
B Subtraction
C Multiplication
D Division
46 What is the inverse operation needed to solve
this equation?
x - 3 = -12
A
B
C
D
Addition
Subtraction
Multiplication
Division
47 What is the inverse operation needed to
solve this problem?
-2 + x = 9
A
B
C
D
Addition
Subtraction
Multiplication
Division
One Step Equations
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To solve equations, you must work backwards through
the order of operations to find the value of the variable.
Remember to use inverse operations in order to isolate
the variable on one side of the equation.
Whatever you do to one side of an equation, you MUST
do to the other side!
Examples:
y + 3 = 13
- 3 -3
y = 10
The inverse of adding 3 is subtracting 3
4m = 32
4
4
m=8
The inverse of multiplying by 4 is dividing by 4
Remember - whatever you do to one side of an equation,
you MUST do to the other!!!
One Step Equations
Solve each equation then click the box to see work &
solution.
2=x-4
x-5=2
+4
+4
+5 +5
click to show
click to show
6
= xoperation
x
=
7
inverse
inverse operation
x + 5 = -14
-5 -5
click to show
xoperation
= -19
inverse
6=x+1
-1click to show
-1
inverse
5 = xoperation
x+9=5
-9 -9
click to show
x =operation
-4
inverse
12 = x + 17
-17
-17
click to show
-5inverse
= x operation
One Step Equations
4x = 16
4
4
click to show
inverse
x =operation
4
-2x = -12
-2
-2
click to show
inverse
x =operation
6
(2) x = 9 (2)
2
x = 18
click to show
inverse operation
(-6)
x = 36 (-6)
-6
x = -216
click to show
inverse operation
-20 = 5x
5
5
click to show
inverse
-4 =operation
x
48
Solve.
x - 7 = 19
49
Solve.
j + 15 = 17
50
Solve.
42 = 6y
51
Solve.
-115 = -5x
52
Solve.
x
9
= 12
53
Solve.
w - 17 = 37
54
Solve.
x
-3 = 7
55
Solve.
23 + t = 11
56
Solve.
108 = 12r
Sometimes the operation can be confusing.
For example: -2 + x = 7
This looks like you should use subtraction to undo the
problem. However, -2 + x = 7 is the same as x - 2 = 7
so while it appears to be addition, it is really subtraction.
In order to undo this we can add.
-2 + x = 7
x-2=7
+2 +2
x=9
OR
-2 + x = 7
- (-2) -(-2)
x=9
OR
-2 + x = 7
+2
+2
x=9
-2 + x = 7
-2
= -2
-4 + x = 5
This did not
cancel out
anything.
-2 + x = 7
+2 +2
x=9
This did
cancel out to
find the
answer.
-2 + x = 7
x-2=7
+2 +2
x=9
This is the
same
as the middle
problem
Try these:
1.) -4 + b = 7
2.) -2 + r = 4
3.) -3 + w = 6
4.) -5 + c = 9
Think about this...
In the expression
To which does the "-" belong?
Does it belong to the x? The 3? Both?
The answer is that there is one negative so it is used
once with either the variable or the 3. Generally, we
assign it to the 3 to avoid creating a negative variable.
So:
57 Solve.
58
Solve.
-5 + q = 15
59
Solve.
60
Solve
61
Solve.
62
Solve.
63
Solve.
Sometimes you will have an equation where you are
multiplying a variable by a fraction.
𝟑
=𝟗
𝟒𝒙
To undo the fraction you:
Multiply by the Reciprocal of the Coefficent
This means that you will flip the fraction and then
multiply
**Dividing by a fraction is the same
as multiplying by its reciprocal
1 times any
number is
itself so this
is why it can
cancel out.
64
Solve.
65
Solve
66
Solve.
Two-Step Equations
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Sometimes it takes more than one step to solve an
equation. Remember that to solve equations, you must
work backwards through the order of operations to find
the value of the variable.
This means that you undo in the opposite order
(PEMDAS):
1st: Addition & Subtraction
2nd: Multiplication & Division
3rd: Exponents
4th: Parentheses
Whatever you do to one side of an equation, you MUST
do to the other side!
Examples:
4x + 2 = 10
-2 -2
4x = 8
4 4
x=2
-2y - 9 = -13
+9 + 9
-2y = -4
-2
-2
y=2
Undo addition first
Undo multiplication second
Undo subtraction first
Undo multiplication second
Remember - whatever you do to one side
of an equation, you MUST do to the other!!!
Two Step Equations
Solve each equation then click the box to see
work & solution.
5b + 3 = 18
3j - 4 = 14
w + 6 = 10
-3 -3
+4 +4
2
5b = 15
3j = 18
-6 -6
5
5
3 3
w 2 = 4 2
b=3
j=6
2
w=8
-2x + 3 = -1
-2m - 4 = 22
- 3 -3
+4 +4
+5 = +5
-2x = -4
-2m = 26
-2 -2
-2 -2
x=2
t = 15
m = -13
67
Solve the equation.
5x - 6 = -56
68
Solve the equation.
14 = 3c + 2
69
Solve the equation.
x - 4 = 24
5
70
Solve the equation.
5r - 2 = -12
71 Solve the equation.
14 = -2n - 6
72
Solve the equation.
x
5 + 7 = 13
73
Solve the equation.
x
+ 2 = -10
3
74
Solve the equation.
75
Solve the equation.
76 Solve the equation.
77 Solve the equation.
78
Solve
-3
1
1
x+
=
5
2 10
79
Solve the equation.
80 Solve the equation.
Multi-Step Equations
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Steps for Solving Multiple Step Equations
As equations become more complex, you should:
1. Simplify each side of the equation.
(Combining like terms and the distributive property)
2. Use inverse operations to solve the equation.
Remember, whatever you do to one side of an equation,
you MUST do to the other side!
Examples:
5x + 7x + 4 = 28
12x + 4 = 28
-4 - 4
12x = 24
12 12
x=2
-1 = 2r - 7r +19
-1 = -5r + 19
-19 =
- 19
-20 = -5r
-5
-5
4=r
Combine Like Terms
Undo Addition
Undo Multiplication
Combine Like Terms
Undo Subtraction
Undo Multiplication
Try these.
12h - 10h + 7 = 25
h=9
-17q + 7q -13 = 27
q=-4
17 - 9f + 6 = 140
f = -13
Always check to see that both sides of the equation are
simplified before you begin solving the equation.
Sometimes, you need to use the distributive property in
order to simplify part of the equation.
Remember: The distributive property is a(b + c) = ab + ac
Examples
5(20 + 6) = 5(20) + 5(6)
9(30 - 2) = 9(30) - 9(2)
3(5 + 2x) = 3(5) + 3(2x)
-2(4x - 7) = -2(4x) - (-2)(7)
Examples:
2(b - 8) = 28
2b - 16 = 28
+16 +16
2b = 44
2
2
b = 22
3r + 4(r - 2) = 13
3r + 4r - 8 = 13
7r - 8 = 13
+8 +8
7r = 21
7
7
r=3
Distribute the 2 through (b - 8)
Undo subtraction
Undo multiplication
Distribute the 4 through (r - 2)
Combine Like Terms
Undo subtraction
Undo multiplication
Try these.
3(w - 2) = 9
w=5
4(2d + 5) = 92
d=9
6m + 2(2m + 7) = 54
m=4
81
Solve.
9 + 3x + x = 25
82
Solve
-8e + 7 +3e = -13
83
Solve.
-27 = 8x - 4 - 2x - 11
84
Solve.
n - 2 + 4n - 5 = 13
85
Solve.
32 = f - 3f + 6f
86
Solve.
6g - 15g + 8 - 19 = -38
87
Solve.
3(a - 5) = -21
88
Solve.
4(x + 3) = 20
89
Solve.
3 = 7(k - 2) + 17
90
Solve.
2(p + 7) -7 = 5
91
Solve.
3m -1m + 3(m-2) = 19.75
92
Solve.
93
Solve.
94
Solve.
Distributing Fractions in
Equations
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of contents
Remember...
1. Simplify each side of the equation.
2. Solve the equation.
(Undo addition and subtraction first, multiplication and
division second)
Remember, whatever you do to one side of an equation,
you MUST do to the other side!
There is more than one way to solve an equation with a
fraction coefficient. While you can, you don't need to
distribute.
3 (-3 + 3x) = 72
5
5
Multiply by the reciprocal
3 (-3 + 3x) = 72
5
5
5  3 (-3 + 3x) = 72  5
3 5
5
3
-3 + 3x = 24
+3
+3
3x = 27
3
3
x=9
Multiply by the LCD
3 (-3 + 3x) = 72
5
5
5  3 (-3 + 3x) = 72  5
5
5
3(-3 + 3x) = 72
-9 + 9x = 72
+9
+9
9x = 81
9
9
x=9
Some problems work better when you multiply by the
reciprocal and some work better multiplying by the LCM.
Which strategy would you use for the following? Why?
95
Solve.
96
Solve.
97
Solve.
2 (8 - 3c) = 16
3
3
98
Solve.
99
Solve.
Translating Between
Words and Expressions
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of contents
List words that indicate
addition
List words that indicate
subtraction
List words that indicate
multiplication
List words that indicate
division
List words that indicate
equals
Be aware of the difference between "less" and "less than".
For example:
"Eight less three" and "Three less than Eight" are
equivalent expressions. So what is the difference in
wording?
Eight less three:
Three less than eight:
8-3
8-3
When you see "less than", you need to switch the order of
the numbers.
As a rule of thumb, if you see the words "than" or
"from" it means you have to reverse the order
of the two items on either side of the word.
Examples:
·8 less than b means _______________
b-8
·3 more than x means x_______________
+3
·x less than 2 means 2_______________
-x
click to reveal
The many ways to represent multiplication...
How do you represent "three times a"?
(3)(a)
3(a)
3 a
3a
The preferred representation is 3a
When a variable is being multiplied by a number, the
number (coefficient) is always written in front of the
variable.
The following are not allowed:
3xa ... The multiplication sign looks like another
variable
a3 ... The number is always written in front of the
variable
Representation of division...
How do you represent "b divided by 12"?
b ÷ 12
b ∕ 12
b
12
When choosing a variable, there are some
that are often avoided:
l, i, t, o, O, s, S
Why might these be avoided?
It is best to avoid using letters that might be
confused for numbers or operations. In the
case above (1, +, 0, 5)
TRANSLATE THE WORDS
INTO AN
ALGEBRAIC EXPRESSION
Three times j
Eight divided by j
+
j
-
j less than 7

5 more than j
4 less than j
÷
0
1
2
3
4
5
6
7
8
9
Write the expression for each statement.
Then check your answer.
The sum of twenty-three and m
23 + m
Write the expression for each statement.
Then check your answer.
Twenty-four less than d
d - 24
Write the expression for each statement.
Remember, sometimes you need to use
parentheses for a quantity.
Four times the difference of eight and j
4(8-j)
Write the expression for each statement.
Then check your answer.
The product of seven and w, divided by 12
7w
12
Write the expression for each statement.
Then check your answer.
The square of the sum of six and p
(6+p)
2
100
The quotient of 200 and the quantity of
p times 7
A
B
200
7p
200 - (7p)
C
200 ÷ 7p
D
7p
200
101
35 multiplied by the quantity r less 45
A
35r - 45
B
35(45) - r
C
35(45 - r)
D
35(r - 45)
102 Mary had 5 jellybeans for each of 4 friends.
A
B
C
D
5+4
5-4
5x4
5÷4
103
If n + 4 represents an odd integer, the next
larger odd integer is represented by
A
B
C
D
n+2
n+3
n+5
n+6
From the New York State Education Department. Office of Assessment Policy, Development
and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra;
accessed 17, June, 2011.
104 a less than 27
A
27 - a
B
C
a
27
a - 27
D
27 + a
105
If h represents a number, which equation
is a correct translation of:
“Sixty more than 9 times a number is 375”?
A
B
C
D
9h = 375
9h + 60 = 375
9h - 60 = 375
60h + 9 = 375
From the New York State Education Department. Office of Assessment Policy, Development
and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra;
accessed 17, June, 2011.
Using Numerical and
Algebraic Expressions and
Equations
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of contents
We can use our algebraic translating skills to solve other
problems.
We can use a variable to show an unknown.
A constant will be any fixed amount.
If there are two separate unknowns, relate one to the other.
The school cafeteria sold 225 chicken meals today. They
sold twice the number of grilled chicken sandwiches
than chicken tenders. How many of each were sold?
2c + c = 225
chicken
sandwiches
chicken
tenders
total
meals
c + 2c = 225
3c = 225
3
3
c = 75
The cafeteria sold
150 grilled chicken
sandwiches and
75 tenders.
Julie is matting a picture in a frame. Her frame is 9 12 inches
wide and her picture is 7 inches wide. How much matting
should she put on either side?
1
1
2m + 7 = 9 2
both sides
of the mat
size of
picture
size of
frame
2m + 7 = 9 2
-7 -7
1
2m = 2 2
2
2
1
m = 14
Julie needs 1 14 inches on
each side.
Many times with equations there will be one number that
will be the same no matter what (constant) and one that
can be changed based on the problem (variable and
coefficient).
Example: George is buying video games online. The cost
of the video is $30.00 per game and shipping is a flat fee of
$7.00. He spent a total of $127.00. How many games did he
buy in all?
George is buying video games online. The cost of the
video is $30.00 per game and shipping is a flat fee of $7.00.
He spent a total of $127.00. How many games did he buy in
all?
Notice that the video games are "per game" so that means
there could be many different amounts of games and
therefore many different prices. This is shown by writing
the amount for one game next to a variable to indicate any
number of games.
30g
cost of
one video
game
number
of
games
George is buying video games online. The cost of the
video is $30.00 per game and shipping is a flat fee of $7.00.
He spent a total of $127.00. How many games did he buy in
all?
Notice also that there is a specific amount that is charged
no matter what, the flat fee. This will not change so it is the
constant and it will be added (or subtracted) from the other
part of the problem.
30g
cost of
one video
game
+7
number
of
games
the cost
of the
shipping
George is buying video games online. The cost of the
video is $30.00 per game and shipping is a flat fee of $7.00.
He spent a total of $127.00. How many games did he buy in
all?
"Total" means equal so here is how to write the rest of the
equation.
30g
cost of
one video
game
+7
number
of games
the cost
of the
shipping
= 127
the total
amount
George is buying video games online. The cost of the
video is $30.00 per game and shipping is a flat fee of $7.00.
He spent a total of $127.00. How many games did he buy in
all?
Now we can solve it.
30g + 7 = 127
-7
-7
30g = 120
30
30
g=4
George bought 4 video games.
106 Lorena has a garden and wants to put a
gate to her fence directly in the middle of
one side. The whole length of the fence is
1
24 feet. If the gate is 4 2 feet, how many feet
should be on either side of the fence?
107 Lewis wants to go to the amusement park
with his family. The cost is $12.00 for parking
plus $27.00 per person to enter the park.
Lewis and his family spent $147. Which
equation shows this problem?
A 12p + 27 = 147
B 12p + 27p = 147
C 27p + 12 = 147
D 39p = 147
108
Lewis wants to go to the amusement park
with his family. The cost is $12.00 for parking
plus $27.00 per person to enter the park.
Lewis and his family spent $147. How many
people went to the amusement park WITH
Lewis?
109
Mary is saving up for a new bicycle that is
$239. She has $68.00 already saved. If she
wants to put away $9.00 per week, how many
weeks will it take to save enough for her
bicycle? Which equation represents the
situation?
A
B
C
D
9 + 68 = 239
9d + 68 = 239
68d + 9 = 239
77d = 239
110
Mary is saving up for a new bicycle that is
$239. She has $68.00 already saved. If she
wants to put away $9.00 per week, how
many weeks will it take to save enough for
her bicycle?
111
You are selling t-shirts for $15 each as a
fundraiser. You sold 17 less today then
you did yesterday. Altogether you have
raised $675.
Write and solve an equation to
determine the number of t-shirts you
sold today.
Be prepared to show your equation!
112
Rachel bought $12.53 worth of school
supplies. She still needs to buy pens which
are $2.49 per pack. She has a total of
$20.00 to spend on school supplies. How
many packs of pens can she buy?
Write and solve an equation to determine
the number of packs of pens Rachel can
buy.
Be prepared to show your equation!
113
The length of a rectangle is 9 cm greater
than its width and its perimeter is 82 cm.
Write and solve an equation to determine
the width of the rectangle.
Be prepared to show your equation!
114
The product of -4 and the sum of 7 more
than a number is -96.
Write and solve an equation to determine
the number. Be prepared to show your
equation!
115
A magazine company has 2,100 more
subscribers this year than last year.
Their magazine sells for $182 per year.
Their combined income from last year
and this year is $2,566,200.
Write and solve an equation to determine
the number of subscribers they had each
year.
Be prepared to show your equation!
How many subscribers last year?
116
A magazine company has 2,100 more
subscribers this year than last year.
Their magazine sells for $182 per year.
Their combined income from last year
and this year is $2,566,200.
Write and solve an equation to determine
the number of subscribers they had each
year.
Be prepared to show your equation!
How many subscribers this year?
117 The perimeter of a hexagon is 13.2 cm.
Write and solve an equation to determine
the length of a side of the hexagon.
Be prepared to show your equation!
Graphing and Writing
Inequalities
with One Variable
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of contents
When you need to use an inequality to solve a word
problem, you may encounter one of the phrases below.
Equivalent
Translation
Important
Words
Sample
Sentence
is more than
Trenton is more
than 10 miles
away.
t > 10
is greater than
A is greater than
B.
A>B
must exceed
The speed must
exceed 25 mph.
The speed is
greater than 25
mph.
s > 25
When you need to use an inequality to solve a word problem, you may encounter one of
the phrases below.
Important
Words
Sample
Sentence
Equivalent
Translation
cannot exceed
Time cannot
exceed 60
minutes.
Time must be
less than or
equal to 60
minutes.
t < 60
is at most
At most, 7
students were
late for class.
Seven or fewer
students were
late for class.
n<7
is at least
Bob is at least
14 years old.
Bob's age is
greater than or
equal to 14.
B > 14
How are these inequalities read?
2+2>3
Two plus two is greater than 3
2+2>3
Two plus two is greater than or equal to 3
2+2≥4
Two plus two is greater than or equal to 4
2+2<5
Two plus two is less than 5
2+2≤5
Two plus two is less than or equal to 5
2+2≤4
Two plus two is less than or equal to 4
Writing inequalities
Let's translate each statement into an inequality.
words
x is less than 10
translate to
x
<
10
inequality statement
20 is greater than or equal to
y
20
y
>
You try a few:
1. 14 is greater than a
2. b is less than or equal to 8
3. 6 is less than the product of f and 20
4. The sum of t and 9 is greater than or equal to 36
5. 7 more than w is less than or equal to 10
6. 19 decreased by p is greater than or equal to 2
7. Fewer than 12 items
8. No more than 50 students
9. At least 275 people attended the play
Do you speak math?
Change the following expressions from English into
math.
Double a number is at most four.
2xAnswer
≤4
Three plus a number is at least six.
3+x≥6
Answer
Five less than a number is less than twice that number.
Answer
x-5<
2x
The sum of two consecutive numbers is at least
thirteen.
x + (xAnswer
+ 1) ≥ 13
Three times a number plus seven is at least nine.
3x + 7 > 9
Answer
A store's employees earn at least $7.50 per hour. Define
a variable and write an inequality for the amount the
employees may earn per hour.
Let e represent an employee's wages.
An employee earns
at least
$7.50
e
>
7.5
7.5
0
1
2
3
4
5
6
7
8
9
10
Try this:
The speed limit on a road is 55 miles per hour. Define a
variable and write an inequality.
118
You have $200 to spend on clothes. You
already spent $140 and shirts cost $12.
Which equation shows this scenario?
A
B
C
D
200 < 12x + 140
200 ≤ 12x + 140
200 > 12x + 140
200 ≥ 12x + 140
119 A sea turtle can live up to 125 years. If one is
already 37 years old, which scenario shows
how many more years could it live?
A
B
C
125 < 37 + x
125 ≤ 37 + x
125 > 37 + x
D
125 ≥ 37 + x
120
The width of a rectangle is 3 in longer than
the length. The perimeter is no less than 25
inches.
A
B
C
D
4a + 6 < 25
4a + 6 ≤ 25
4a + 6 > 25
4a + 6 ≥ 25
121 The absolute value of the sum of two
numbers is less than or equal to the sum of
the absolute values of the same two numbers.
A
B
C
D
Solution Sets
A solution to an inequality is NOT a single
number. It will have more than one value.
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
This would be read as the solution set is all
numbers greater than or equal to negative 5.
Let's name the numbers that are solutions of the given
inequality.
r > 10
Which of the following are solutions? {5, 10, 15, 20}
5 > 10 is not true
So, not a solution
10 > 10 is not true
So, not a solution
15 > 10 is true
So, 15 is a solution
20 > 10 is true
So, 20 is a solution
Answer:
{15, 20} are solutions of the inequality r > 10
Let's try another one.
30 ≥ 4d; {3, 4, 5, 6, 7, 8}
30 ≥ 4d
30 ≥ (4)3
reveal
30click≥to12
30 ≥ 4d
30 ≥ (4)4
reveal
30click
≥ to16
30 ≥ 4d
30 ≥ (4)5
reveal
30click
≥ to20
30 ≥ 4d
30 ≥ (4) 6
30click≥ to24
reveal
30 ≥ 4d
30 ≥ (4)7
30click
≥ to28
reveal
30 ≥ 4d
30 ≥ (4)8
30click
≥ to32
reveal
Graphing Inequalities - The Circle
An open circle on a number shows that the number
is not part of the solution. It is used with "greater
than" and "less than". The word equal is not
included.< >
A closed circle on a number shows that the number
is part of the solution. It is used with "greater than
or equal to" and "less than or equal to". < >
Graphing Inequalities - The Arrow
The arrow should always point in the direction of those
numbers who satisfy the inequality.
*If the variable is on the left side of the inequality, then
< and ≤ will show an arrow pointing left.
*If the variable is on the left side of the inequality, then
> and ≥ will show an arrow pointing right.
Notice that < and ≤ look like an arrow pointing left
and that > and ≥ look like an arrow pointing right.
But what if the variable isn't on the left?
Do the opposite of where the inequality symbol points.
-5
-4
-3
-2
-1
0
1
2
3
4
5
Graphing Inequalities
What is the number in the inequality?
What kind of circle should be used?
In what direction does the line go?
Graphing Inequalities
x is less than 5
Step 1: Rewrite this as x < 5.
Step 2: What kind of circle? Because it is less than, it
does not include the number 5 and so it is an open
circle.
-5
-4
-3
-2
-1
0
1
2
3
4
5
x<5
Step 3: Draw an arrow on the number line showing all
possible solutions. Numbers greater than the variable, go
to the right. Numbers less than the variable, go to the left.
-5
-4
-3
-2
-1
1
0
2
3
5
4
Step 4: Draw a line, thicker than the horizontal line, from the
dot to the arrow. This represents all of the numbers that
fulfill the inequality.
-5
-4
-3
-2
-1
0
1
2
3
4
5
Graphing Inequalities
x is less than or equal to 5
Step 1: Rewrite this as x ≤ 5.
Step 2: What kind of circle? Because it is less than or
equal to, it does include the number 5 and so it is a
closed circle.
-5
-4
-3
-2
-1
0
1
2
3
4
5
x≤5
Step 3: Draw an arrow on the number line showing all
possible solutions. Numbers greater than the variable, go
to the right. Numbers less than the variable, go to the left.
-5
-4
-3
-2
-1
1
0
2
3
5
4
Step 4: Draw a line, thicker than the horizontal line, from the
dot to the arrow. This represents all of the numbers that
fulfill the inequality.
-5
-4
-3
-2
-1
0
1
2
3
4
5
You try
Graph the inequality
x>2
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
.
0
5
.
02
5
3
4
5
6
7
8
9
10
click 2 on the
number line for
answer
Graph the inequality
-3 > x
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
click -3 on the
number line for
answer
Try these.
Graph the inequalities.
1. x > -3
-5
-4
.
0
5
-3
.
-2
-1
0
1
2
3
4
5
-3
-2
-1
0
1
2
3
4
5
2. x < 4
-5
-4
Try these.
State the inequality shown.
1.
-5
-4
-3
-2
-1
0
1
2
3
4
5
2.
-5
-4
-3
-2
-1
0
1
2
3
4
5
122 This solution set would be x > -4.
A
B
-10 -9
-8
-7
-6
True
False
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
123 State the inequality shown.
-5
-4
-3
-2
-1
A
x>3
B
x<3
C
x<3
D
x>3
0
1
2
3
4
5
124
State the inequality shown.
5
6
7
8
9
A
11 < x
B
11 > x
C
11 > x
D
11 < x
10
11 12 13
14 15
125
State the inequality shown.
-5
-4
-3
-2
-1
A
x > -1
B
x < -1
C
x < -1
D
x > -1
0
1
2
3
4
5
126
State the inequality shown.
-5
-4
-3
-2
-1
A
-4 < x
B
-4 > x
C
-4 < x
D
-4 > x
0
1
2
3
4
5
127
State the inequality shown.
-5
-4
-3
-2
A
x>0
B
x<0
C
x<0
D
x>0
-1
0
1
2
3
4
5
Simple Inequalities
Involving Addition
and Subtraction
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of contents
Remembers how to solve an algebraic equation?
x + 3 = 13
-3 -3
x = 10
Use the inverse of addition
Does 10 + 3 = 13
Be sure to13
check
= 13 your
answer!
· Solving one-step inequalities is much like solving
one-step equations.
·To solve an inequality, you need to isolate the
variable using the properties of inequalities and
inverse operations.
· Remember, whatever you do to one side, you do to
the other.
To find the solution, isolate the variable x.
Remember, it is isolated when it appears by itself on
one side of the equation.
12 > x + 5
-5
-5
7>x
Subtract to undo addition
7>x
The symbol is > so it is an open circle and it is numbers
less than 7 so it goes to the left.
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
Solve and graph.
A. j + 7 > -2
A. j + 7 > -2
-7 -7
j > -9
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-9 is not included in solution set; therefore we
graph with an open circle.
Solve and graph.
B. r - 2 > 4
r-2>4
+2 +2
r> 6
0
1
2
3
4
5
6
7
8
9
10 11
12 13 14
Solve and graph.
9>w+4
-4
-4
C. 9 > w + 4
5>w
w<5
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
128 Solve the inequality.
3<s+4
____ < s
129
Solve the inequality and graph the solution.
-4 + b < -2
A
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
B
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
C
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-4
-3
6
7
D
-10
-9
-8
-7
-6
-5
-2
-1
0
1
2
3
4
5
8
9
10
130 Solve the inequality and graph the solution.
-8 > b - 5
A
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
B
C
D
131
Solve the inequality.
m + 6.4 < 9.6
m < ______
Simple Inequalities
Involving Multiplication
and Division
Return to table
of contents
Multiplying or Dividing by a Positive
Number
3x > -36
3x > -36
3
3
x > -12
Since x is multiplied by 3, divide
both sides by 3 for the inverse
operation.
Solve the inequality.
2 r < 4
3
( )
3
2
2 r < 4
3
r < 6
( )
3
2
Since r is multiplied by 2/3,
multiply both sides by the
reciprocal of 2/3.
132
3k > 18
A
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
B
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
C
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
D
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
133
-30 > 3q
A
10 > q
B
-10 < q
C
-10 > q
D
10 < q
134
X
< -3
2
A
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
B
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
C
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
D
135
3 g > 27
4
A
g > 36
B
g > 108
C
g > 36
D
g > 108
136
-21 > 3d
A
d > -7
B
d > -7
C
d < -7
D
d < -7
Now let's see what happens when we multiply or
divide by negative numbers.
·Sometimes you must multiply or divide to isolate
the variable.
·Multiplying or dividing both sides of an inequality
by a negative number gives a surprising result.
1. Write down two numbers and put the appropriate
inequality (< or >) between them.
2. Apply each rule to your original two numbers from
step 1 and simplify. Write the correct inequality
(< or >) between the answers.
A. Add 4
B. Subtract 4
C. Multiply by 4
D. Multiply by -5
E. Divide by 4
F. Divide by -4
3. What happened with the inequality symbol in your
results?
4. Compare your results with the rest of the class.
5. What pattern(s) do you notice in the inequalities?
How do different operations affect inequalities?
Write a rule for inequalities.
Let's see what happens when we multiply this inequality
by -1.
5 > -1
-1 • 5 ? -1 • -1
We know 5 is greater than -1
Multiply both sides by -1
-5 ? 1
Is -5 less than or greater than 1?
-5 < 1
You know -5 is less than 1, so you
should use <
What happened to the inequality symbol to keep the
inequality statement true?
Helpful Hint
The direction of the inequality changes only if the
number you are using to multiply or divide by is
negative.
Solve and graph.
A. -3y > 18
Dividing each side by -3 changes
the > to <.
-3y < 18
-3
-3
y < -6
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
Solve and graph.
B.
-7m > -28
-7m < -28
-7
-7
m< 4
-10 -9
-8
-7
-6
Divide each side by -7
Flip the sign because you
divided by a negative.
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
Solve and graph.
C. 5m > -25
5m > -25
5
5
Divide each side by 5.
m > -5
-10 -9
-8
The sign does NOT change
because you did not divide by
a negative.
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
Solve and graph.
D. -8y > 32
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-10
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
E. -9f > -54
-9
-r
< 5
2
(-2)
-r
> 5
2
( )
Multiply both sides by the
reciprocal of -1/2.
-2
Why did the inequality change?
r > -10
You multiplied by a negative.
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
Try these.
Solve and graph each inequality.
1. -6h < 42
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
2. 4x > -20
Try these.
Solve and graph each inequality.
3. 5m < 30
4.
a > -3
-2
-10
-9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-10
-9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
137
Solve and graph.
3y < -6
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
138
Solve and graph.
x < -2
-4
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
139
Solve and graph.
-5y ≤ -25
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
140
Solve and graph.
n>2
-2
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10