Setting the PowerPoint View Use Normal View for the Interactive
Elements To use the interactive elements in this presentation, do
not select the Slide Show view. Instead, select Normal view and
follow these steps to set the view as large as possible: On the
View menu, select Normal. Close the Slides tab on the left. In the
upper right corner next to the Help button, click the ^ to minimize
the ribbon at the top of the screen. On the View menu, confirm that
Ruler is deselected. On the View tab, click Fit to Window. On the
View tab, click Slide Master | Page Setup. Select On-screen Show
(4:3) under Slide sized for and click Close Master View. On the
Slide Show menu, confirm that Resolution is set to 1024x768. Use
Slide Show View to Administer Assessment Items To administer the
numbered assessment items in this presentation, use the Slide Show
view. (See Slide 10 for an example.)
Slide 3
Table of Contents Inverse Operations One Step Equations Two
Step Equations Multi-Step Equations Distributing Fractions in
Equations Graphing & Writing Inequalities with One Variable
Click on a topic to go to that section. The Distributive Property
and Factoring Combining Like Terms Simple Inequalities involving
Addition & Subtraction Simple Inequalities involving
Multiplication & Division Common Core Standards: 7.EE.1,
7.EE.3, 7.EE.4 Simplifying Algebraic Expressions Translating
Between Words and Equations Commutative and Associative Properties
Using Numerical and Algebraic Expressions and Equations
Slide 4
Commutative and Associative Properties Return to table of
contents
Slide 5
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)
Slide 6
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
Slide 7
The Commutative Property is particularly useful when you are
combining integers. Example: -15 + 9 + (-4)= -15 + (-4) +
9=Changing it this way allows for the -19 + 9 = negatives to be
added together first. -10
Slide 8
Associative Property of Multiplication: The order in which the
terms of a product are grouped does not change the product.
Slide 9
1Identify the property of -5 + 3 = 3 + (-5) ACommutative
Property of Addition BCommutative Property of Multiplication
CAssociative Property of Addition DAssociative Property of
Multiplication
Slide 10
2Identify the property of a + (b + c) = (a + c) + b
ACommutative Property of Addition BCommutative Property of
Multiplication CAssociative Property of Addition DAssociative
Property of Multiplication
Slide 11
3 Identify the property of (3 * (-4)) * 8 = 3 * ((-4) * 8)
ACommutative Property of Addition BCommutative Property of
Multiplication CAssociative Property of Addition DAsociative
Property of Multiplication
Slide 12
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)
Slide 13
Combining Like Terms Return to table of contents
Slide 14
An Algebraic Expression - contains numbers, variables and at
least one operation.
Slide 15
Like terms: terms in an expression that have the same variable
raised to the same power Examples: LIKE TERMS NOT LIKE TERMS 6x and
2x 6x 2 and 2x 5y and 8y 5x and 8y 4x 2 and 7x 2 4x 2 y and 7xy
2
Slide 16
4 Identify all of the terms like 2x A5x B3x 2 C5y D12y E2
Slide 17
5 Identify all of the terms like 8y A9y B4y 2 C7y D8 E-18x
Slide 18
6 Identify all of the terms like 8xy A8x B3x 2 y C39xy D4y
E-8xy
Slide 19
7 Identify all of the terms like 2y A51w B2x C3y D2w E-10y
Slide 20
8 Identify all of the terms like 14x 2 A-5x B8x 2 C13y 2 Dx E-x
2
Slide 21
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
Slide 22
Sometimes there are constant terms that can be combined. 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.
The Distributive Property and Factoring Return to table of
contents
Slide 36
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. 4 7 3 How could you express the area of those two rooms
together?
Slide 37
4 7 +3 Either way, the area is 40 feet 2 : You could add 7 + 3
and then multiply by 4 4(7+3)= 4(10)= 40 OR You could multiply 4 by
7, then 4 by 3 and add them 4(7) + 4(3) = 28 + 12 = 40 4 3 7 4
Slide 38
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. 4 x 3 How could you express the area of
those two rooms together?
Slide 39
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) + 4(3)= 4x + 12 The area of the two rooms is 4x + 12 (Note: 4x
cannot be combined with 12)
Slide 40
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)
Slide 41
Write an expression equivalent to: 5(y + 4) 5(y) + 5(4) 5y + 20
6(x + 2)3(x + 4) 4(x - 5) 7(x - 1) Remember to distribute the 5 to
the y and the 4
Slide 42
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. EXAMPLE: -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) = Be careful with your signs!
Slide 43
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? The numbers are turned to their
opposites. Remove to see answer. Try these: -(9x + 3) = -(-5x + 1)
= -(2x - 4) = -(-x - 6) =
Slide 44
20 4(2 + 5) = 4(2) + 5 A True B False
Slide 45
21 8(x + 9) = 8(x) + 8(9) A True B False
Slide 46
22 -4(x + 6) = -4 + 4(6) A True B False
Slide 47
23 3(x - 4) = 3(x) - 3(4) A True B False
Slide 48
24Use the distributive property to rewrite the expression
without parentheses 3(x + 4) A3x + 4 B3x + 12 Cx + 12 D7x
Slide 49
25Use the distributive property to rewrite the expression
without parentheses 5(x + 7) Ax + 35 B5x + 7 C5x + 35 D40x
Slide 50
26Use the distributive property to rewrite the expression
without parentheses (x + 5)2 A2x + 5 B2x + 10 Cx + 10 D12x
Slide 51
27Use the distributive property to rewrite the expression
without parentheses 3(x - 4) A3x - 4 Bx - 12 C3x - 12 D9x
Slide 52
28Use the distributive property to rewrite the expression
without parentheses 2(w - 6) A2w - 6 Bw - 12 C2w - 12 D10w
Slide 53
29Use the distributive property to rewrite the expression
without parentheses -4(x - 9) A-4x - 36 Bx - 36 C4x - 36 D-4x +
36
Slide 54
30Use the distributive property to rewrite the expression
without parentheses 5.2(x - 9.3) A-5.2x - 48.36 B5.2x - 48.36
C-5.2x + 48.36 D-48.36x
Slide 55
31Use the distributive property to rewrite the expression
without parentheses A B C D
Slide 56
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 + 35Both 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.
Slide 57
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)
-What divides into the expression: -5n - 20mn - 10np
Slide 65
-If a regular pentagon has a perimeter of 10x + 25, what does
each side equal?
Slide 66
Simplifying Algebraic Expressions Return to table of
contents
Slide 67
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.
Slide 68
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.
Slide 69
37 7x +3(x - 4) = 10x - 4 A True B False
Slide 70
38 8 +(x + 3)5 = 5x + 11 A True B False
Slide 71
39 4 +(x - 3)6 = 6x -14 A True B False
Slide 72
40 2x + 3y + 5x + 12 = 10xy + 12 A True B False
Slide 73
41 5x 2 + 2x + 7(x + 1) + x 2 = 6x 2 + 9x + 7 A True B
False
Slide 74
42 2x 3 + 4x 2 + 6(x 2 + 3x) + x = 2x 3 + 10x 2 + 4x A True B
False
Slide 75
43The lengths of the sides of home plate in a baseball field
are represented by the expressions in the accompanying figure.
A5xyz Bx 2 + y 3 z C2x + 3yz D2x + 2y + yz yz y y x x 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.
Which expression represents the perimeter of the figure?
Slide 76
44A rectangle has a width of x and a length that is double
that. What is the perimeter of the rectangle? A 4x B6x C8x
D10x
Slide 77
Inverse Operations Return to table of contents
Slide 78
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
Slide 79
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
Slide 80
An equation can be compared to a balanced scale. Both sides
need to contain the same quantity in order for it to be
"balanced".
Slide 81
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
Slide 82
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).
Slide 83
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
Subtraction Multiplication Division Square Square Root
Slide 84
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.)?
Slide 85
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.
Slide 86
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
Slide 87
For each equation, write the inverse operation needed to solve
for the variable. a.) y +7 = 14 subtract 7 b.) a - 21 = 10 add 21
c.) 5s = 25 divide by 5 d.) x = 5 multiply by 12 12 move
Slide 88
Think about this... To solve c - 3 = 12 Which method is better?
Why? Kendra Added 3 to each side of the equation c - 3 = 12 +3 +3 c
= 15 Ted Subtracted 12 from each side, then added 15. c - 3 = 12
-12 -12 c - 15 = 0 +15 +15 c = 15
Slide 89
45What is the inverse operation needed to solve this equation?
2x = 14 AAddition BSubtraction CMultiplication DDivision
Slide 90
46What is the inverse operation needed to solve this equation?
x - 3 = -12 AAddition BSubtraction CMultiplication DDivision
Slide 91
47What is the inverse operation needed to solve this problem?
-2 + x = 9 AAddition BSubtraction CMultiplication DDivision
Slide 92
One Step Equations Return to table of contents
Slide 93
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!
Slide 94
Examples: y + 3 = 13 - 3 -3 The inverse of adding 3 is
subtracting 3 y = 10 4m = 32 4 4 The inverse of multiplying by 4 is
dividing by 4 m = 8 Remember - whatever you do to one side of an
equation, you MUST do to the other!!!
Slide 95
x - 5 = 2 +5 +5 x = 7 x + 5 = -14 -5 -5 x = -19 2 = x - 4 +4 6
= x 6 = x + 1 5 = x 12 = x + 17 -17 -5 = x x + 9 = 5 -9 -9 x = -4
One Step Equations Solve each equation then click the box to see
work & solution. click to show inverse operation click to show
inverse operation click to show inverse operation click to show
inverse operation click to show inverse operation click to show
inverse operation
Slide 96
One Step Equations 4x = 16 4 4 x = 4 -2x = -12 -2 -2 x = 6 -20
= 5x 5 5 -4 = x x 2 x = 18 = 9 (2) x -6 x = -216 = 36 (-6) click to
show inverse operation click to show inverse operation click to
show inverse operation click to show inverse operation click to
show inverse operation
Slide 97
48 Solve. x - 7 = 19
Slide 98
49 Solve. j + 15 = 17
Slide 99
50 Solve. 42 = 6y
Slide 100
51 Solve. -115 = -5x
Slide 101
52 Solve. = 12 x 9
Slide 102
53 Solve. w - 17 = 37
Slide 103
54 Solve. -3 = x 7
Slide 104
55 Solve. 23 + t = 11
Slide 105
56 Solve. 108 = 12r
Slide 106
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 = 9OR-2 + x = 7 - (-2) -(-2)+2 +2
x = 9
Slide 107
-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
Slide 108
Try these: 1.) -4 + b = 7 2.) -2 + r = 4 3.) -3 + w = 6 4.) -5
+ c = 9
Slide 109
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:
Slide 110
57 Solve.
Slide 111
58 Solve. -5 + q = 15
Slide 112
59 Solve.
Slide 113
60 Solve
Slide 114
61 Solve.
Slide 115
62 Solve.
Slide 116
63 Solve.
Slide 117
Sometimes you will have an equation where you are multiplying a
variable by a fraction.
Slide 118
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
Slide 119
1 times any number is itself so this is why it can cancel
out.
Slide 120
64 Solve.
Slide 121
65 Solve
Slide 122
66 Solve.
Slide 123
Two-Step Equations Return to table of contents
Slide 124
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!
Slide 125
Examples: 4x + 2 = 10 - 2 - 2 Undo addition first 4x = 8 4 4
Undo multiplication second x = 2 -2y - 9 = -13 + 9 + 9 Undo
subtraction first -2y = -4 -2 -2 Undo multiplication second y = 2
Remember - whatever you do to one side of an equation, you MUST do
to the other!!!
Slide 126
5b + 3 = 18 -3 -3 5b = 15 5 5 b = 3 3j - 4 = 14 +4 +4 3j = 18 3
3 j = 6 -2x + 3 = -1 - 3 -3 -2x = -4 -2 -2 x = 2 Two Step Equations
Solve each equation then click the box to see work & solution.
-2m - 4 = 22 +4 +4 -2m = 26 -2 -2 m = -13 +5 = +5 t = 15 w + 6 = 10
2 -6 -6 w 2 = 4 2 2 w = 8
Slide 127
67 Solve the equation. 5x - 6 = -56
Slide 128
68 Solve the equation. 14 = 3c + 2
Slide 129
69 Solve the equation. x 5 - 4 = 24
Slide 130
70 Solve the equation. 5r - 2 = -12
Slide 131
71Solve the equation. 14 = -2n - 6
Slide 132
72 Solve the equation. + 7 = 13 x 5
Slide 133
73 Solve the equation. + 2 = -10 x 3 -
Slide 134
74 Solve the equation.
Slide 135
75 Solve the equation.
Slide 136
76Solve the equation.
Slide 137
77Solve the equation.
Slide 138
78 Solve -3 5 1 2 x + = 1 10
Slide 139
79 Solve the equation.
Slide 140
80Solve the equation.
Slide 141
Multi-Step Equations Return to table of contents
Slide 142
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!
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)
Try these. 3(w - 2) = 9 4(2d + 5) = 92 6m + 2(2m + 7) = 54 w =
5 d = 9 m = 4
Slide 148
81 Solve. 9 + 3x + x = 25
Slide 149
82 Solve -8e + 7 +3e = -13
Slide 150
83 Solve. -27 = 8x - 4 - 2x - 11
Slide 151
84 Solve. n - 2 + 4n - 5 = 13
Slide 152
85 Solve. 32 = f - 3f + 6f
Slide 153
86 Solve. 6g - 15g + 8 - 19 = -38
Slide 154
87 Solve. 3(a - 5) = -21
Slide 155
88 Solve. 4(x + 3) = 20
Slide 156
89Solve. 3 = 7(k - 2) + 17
Slide 157
90 Solve. 2(p + 7) -7 = 5
Slide 158
91 Solve. 3m -1m + 3(m-2) = 19.75
Slide 159
92 Solve.
Slide 160
93 Solve.
Slide 161
94 Solve.
Slide 162
Distributing Fractions in Equations Return to table of
contents
Slide 163
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!
Slide 164
There is more than one way to solve an equation with a fraction
coefficient. While you can, you don't need to distribute. Multiply
by the reciprocal Multiply by the LCD (-3 + 3x) = 3 5 72 5 (-3 +
3x) = 3 5 72 5 (-3 + 3x) = 3535 72 5 3 5 3 -3 + 3x = 24 +3 3x = 27
3 3 x = 9 (-3 + 3x) = 3 5 72 5 (-3 + 3x) = 3 5 72 5 5 5 3(-3 + 3x)
= 72 -9 + 9x = 72 +9 +9 9x = 81 9 9 x = 9
Slide 165
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?
Slide 166
95 Solve.
Slide 167
96 Solve.
Slide 168
(8 - 3c) = 2 3 16 3 97 Solve.
Slide 169
98 Solve.
Slide 170
99 Solve.
Slide 171
Translating Between Words and Expressions Return to table of
contents
Slide 172
List words that indicate addition
Slide 173
List words that indicate subtraction
Slide 174
List words that indicate multiplication
Slide 175
List words that indicate division
Slide 176
List words that indicate equals
Slide 177
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: 8 - 3 Three less than eight: 8 - 3 When you see "less
than", you need to switch the order of the numbers.
Slide 178
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 _______________
Slide 179
The many ways to represent multiplication... How do you
represent "three times a"? (3)(a)3(a) 3 a3a 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
Slide 180
Representation of division... How do you represent "b divided
by 12"? b 12 b 12 b 12
Slide 181
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)
Slide 182
Three times j Eight divided by j j less than 7 5 more than j 4
less than j 1 2 3 4 5 6 7 8 9 0 + - TRANSLATE THE WORDS INTO AN
ALGEBRAIC EXPRESSION j - - - ++
Slide 183
23 + m The sum of twenty-three and m Write the expression for
each statement. Then check your answer.
Slide 184
d - 24 Twenty-four less than d Write the expression for each
statement. Then check your answer.
Slide 185
4(8-j) Write the expression for each statement. Remember,
sometimes you need to use parentheses for a quantity. Four times
the difference of eight and j
Slide 186
7w 12 The product of seven and w, divided by 12 Write the
expression for each statement. Then check your answer.
Slide 187
(6+p) 2 Write the expression for each statement. Then check
your answer. The square of the sum of six and p
Slide 188
100The quotient of 200 and the quantity of p times 7 A200 7p
B200 - (7p) C200 7p D7p 200
Slide 189
101 35 multiplied by the quantity r less 45 A35r - 45 B35(45) -
r C35(45 - r) D35(r - 45)
Slide 190
102Mary had 5 jellybeans for each of 4 friends. A5 + 4 B5 - 4
C5 x 4 D5 4
Slide 191
103If n + 4 represents an odd integer, the next larger odd
integer is represented by An + 2 Bn + 3 Cn + 5 Dn + 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.
Slide 192
104 a less than 27 A27 - a B a 27 Ca - 27 D27 + a
Slide 193
105If h represents a number, which equation is a correct
translation of: Sixty more than 9 times a number is 375? A9h = 375
B9h + 60 = 375 C9h - 60 = 375 D60h + 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.
Slide 194
Using Numerical and Algebraic Expressions and Equations Return
to table of contents
Slide 195
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.
Slide 196
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.
Slide 197
Julie is matting a picture in a frame. Her frame is 9 inches
wide and her picture is 7 inches wide. How much matting should she
put on either side? 2m + 7 = 9 -7 -7 2m = 2 2 2 m = 1 Julie needs 1
inches on each side. 1414 1212 1212 1414 9 both sides of the mat
size of picture size of frame 1212 1212
Slide 198
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?
Slide 199
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
Slide 200
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 + 7
cost of one video game number of games the cost of the
shipping
Slide 201
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 + 7 =
127 cost of one video game number of games the total amount the
cost of the shipping
Slide 202
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 30g = 120 30 30 g = 4George bought 4
video games.
Slide 203
106Lorena 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 24 feet. If the gate is 4 feet, how many feet should be on
either side of the fence? 1212
Slide 204
107Lewis 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? A12p + 27 = 147 B12p + 27p = 147 C27p + 12 = 147 D39p =
147
Slide 205
108Lewis 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?
Slide 206
109Mary 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? A9 + 68 = 239 B9d + 68 = 239
C68d + 9 = 239 D77d = 239
Slide 207
110Mary 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?
Slide 208
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!
Slide 209
112Rachel 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!
Slide 210
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!
Slide 211
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!
Slide 212
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?
Slide 213
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?
Slide 214
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!
Slide 215
Graphing and Writing Inequalities with One Variable Return to
table of contents
Slide 216
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 is more thanTrenton is more than 10
miles away. t > 10 is greater thanA is greater than B. A > B
must exceedThe speed must exceed 25 mph. The speed is greater than
25 mph. s > 25
Slide 217
When you need to use an inequality to solve a word problem, you
may encounter one of the phrases below. Important Words Sample
Sentence EquivalentTranslation cannot exceedTime cannot exceed 60
minutes. Time must be less than or equal to 60 minutes. t < 60
is at mostAt most, 7 students were late for class. Seven or fewer
students were late for class. n < 7 is at leastBob is at least
14 years old. Bob's age is greater than or equal to 14. B >
14
Slide 218
How are these inequalities read? 2 + 2 > 3 Two plus two is
greater than 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
2 + 2 > 3 Two plus two is greater than or equal to 3
Slide 219
Writing inequalities Let's translate each statement into an
inequality. x is less than 10 20 is greater than or equal to y x
< 10 words inequality statement translate to 20 > y
Slide 220
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
Slide 221
Do you speak math? Change the following expressions from
English into math. Double a number is at most four. Three plus a
number is at least six. 2x 4 3 + x 6 Answer
Slide 222
Five less than a number is less than twice that number. The sum
of two consecutive numbers is at least thirteen. Three times a
number plus seven is at least nine. x - 5 < 2x x + (x + 1) 13 3x
+ 7 > 9 Answer
Slide 223
0 1 2 3 4 5 6 7 8 9 10 7.5 $7.50 7.5 at least > An employee
earns e 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.
Slide 224
Try this: The speed limit on a road is 55 miles per hour.
Define a variable and write an inequality.
Slide 225
118You have $200 to spend on clothes. You already spent $140
and shirts cost $12. Which equation shows this scenario? A200 <
12x + 140 B200 12x + 140 C200 > 12x + 140 D200 12x + 140
Slide 226
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?
125 < 37 + x 125 37 + x A B C125 > 37 + x D 125 37 + x
Slide 227
120The width of a rectangle is 3 in longer than the length. The
perimeter is no less than 25 inches. A4a + 6 < 25 B4a + 6 25 C4a
+ 6 > 25 D 4a + 6 25
Slide 228
121The 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
Slide 229
A solution to an inequality is NOT a single number. It will
have more than one value. 1023 4 567 89 10 -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. Solution Sets
Slide 230
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
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".
Slide 233
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.
Slide 234
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. 0 -2-3-4 -5 12 34 5
Slide 235
What is the number in the inequality? What kind of circle
should be used? In what direction does the line go? Graphing
Inequalities
Slide 236
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. 0 -2-3-4 -5 12 34 5 Graphing Inequalities x is
less than 5
Slide 237
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. 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. 0 -2-3-4
-5 12 34 5 x < 5 0 -2-3-4 -5 12 34 5
Slide 238
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. 0 -2-3-4 -5 12 34 5 Graphing
Inequalities x is less than or equal to 5
Slide 239
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. 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. 0 -2-3-4
-5 12 34 5 0 -2-3-4 -5 12 34 5 x 5
Slide 240
10234567 89 10 -2-3-4-5-6-7-8-9-10 You try Graph the inequality
x > 2 Graph the inequality -3 > x 10234567 89 10
-2-3-4-5-6-7-8-9-10 click 2 on the number line for answer click -3
on the number line for answer.05.05.05.05
Slide 241
Try these. Graph the inequalities. 1. x > -3 0 -2 -3 -4-512
34 5 2. x < 4 0 -2-3-4-512 34 5.05..05.
Slide 242
Try these. State the inequality shown. 1. 0 -2-3-4-512 34 5 0
-2-3-4-512 34 5 2.
Slide 243
122 This solution set would be x > -4. 10234567 89 10
-2-3-4-5-6-7-8-9-10 A True B False
Slide 244
0-2-3-4 -5 1234 5 123 Ax > 3 Bx < 3 C Dx > 3 State the
inequality shown.
Slide 245
5 6 7 8 9 10 11 12 13 14 15 124 A11 < x B11 > x C D11
< x State the inequality shown.
Slide 246
0-2-3-4 -5 1234 5 125 Ax > -1 Bx < -1 C Dx > -1 State
the inequality shown.
Slide 247
-5 1 5 0-2-3-4234 126 A-4 < x B-4 > x C-4 < x D-4 >
x State the inequality shown.
Slide 248
0-2-3-4 -5 1234 5 127 Ax > 0 Bx < 0 C Dx > 0 State the
inequality shown.
Slide 249
Simple Inequalities Involving Addition and Subtraction Return
to table of contents
Slide 250
x + 3 = 13 - 3 - 3 x = 10 Remembers how to solve an algebraic
equation? Does 10 + 3 = 13 13 = 13 Be sure to check your answer!
Use the inverse of addition
Slide 251
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.
Slide 252
12 > x + 5 -5 -5 Subtract to undo addition 7 > x To find
the solution, isolate the variable x. Remember, it is isolated when
it appears by itself on one side of the equation.
Slide 253
7 > x The symbol is > so it is an open circle and it is
numbers less than 7 so it goes to the left. 10234567 89 10
-2-3-4-5-6-7-8-9-10
Slide 254
10234567 89 10 -2-3-4-5-6-7-8-9-10 A. j + 7 > -2 Solve and
graph. -9 is not included in solution set; therefore we graph with
an open circle. A. j + 7 > -2 -7 -7 j > -9
Slide 255
B. r - 2 > 4 Solve and graph. 11101213149876543210 r - 2
> 4 +2 +2 r > 6
Slide 256
10234567 89 10 -2-3-4-5-6-7-8-9-10 5 > w - 4 9 > w + 4 w
< 5 C. 9 > w + 4 Solve and graph.
Slide 257
128 Solve the inequality. 3 < s + 4 ____ < s
Slide 258
102 34567 89 10 -2 -3-4-5-6-7-8-9 -10 A 102 34567 89 10 -2
-3-4-5-6-7-8-9 -10 B 102 34567 89 10 -2 -3-4-5-6-7-8-9 -10 C 102
34567 89 10 -2 -3-4-5-6-7-8-9 -10 D 129 Solve the inequality and
graph the solution. -4 + b < -2
Slide 259
102 34567 89 10 -2 -3-4-5-6-7-8-9 -10 A 102 34567 89 10 -2
-3-4-5-6-7-8-9 -10 B 102 34567 89 10 -2 -3-4-5-6-7-8-9 -10 C 102
34567 89 10 -2 -3-4-5-6-7-8-9 -10 D 130 Solve the inequality and
graph the solution. -8 > b - 5
Slide 260
131Solve the inequality. m + 6.4 < 9.6 m < ______
Slide 261
Simple Inequalities Involving Multiplication and Division
Return to table of contents
Slide 262
Since x is multiplied by 3, divide both sides by 3 for the
inverse operation. Multiplying or Dividing by a Positive Number 3x
> -36 3 3 x > -12
Slide 263
Solve the inequality. 2 3 r < 4 3 2 () r < 6 Since r is
multiplied by 2/3, multiply both sides by the reciprocal of 2/3. 2
3 r < 4 3 2 ( )
Slide 264
132 3k > 18 10234567 89 10 -2-3-4-5-6-7-8-9-10 A B C
10234567 89 10 -2-3-4-5-6-7-8-9-10 D 10234567 89 10
-2-3-4-5-6-7-8-9-1010234567 89 10 -2-3-4-5-6-7-8-9-10
Slide 265
133 A B C -30 > 3q 10 > q -10 < q -10 > q D 10 <
q
Slide 266
134 X 2 A B C 102 34567 89 10 -2 -3-4-5-6-7-8-9 -10 D 102 34567
89 10 -2 -3-4-5-6-7-8-9 -10102 34567 89 10 -2 -3-4-5-6-7-8-9 -10102
34567 89 10 -2 -3-4-5-6-7-8-9 -10 < -3
Slide 267
135 A B C g > 27 g > 36 g > 108 g > 36 D g > 108
3 4
Slide 268
136 A B C -21 > 3d d > -7 d < -7 D
Slide 269
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. Now let's see what happens when
we multiply or divide by negative numbers.
Slide 270
1.Write down two numbers and put the appropriate inequality ( )
between them. 2.Apply each rule to your original two numbers from
step 1 and simplify. Write the correct inequality ( ) 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
Slide 271
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.
Slide 272
Let's see what happens when we multiply this inequality by -1.
5 > -1 -1 5 ? -1 -1 -5 ? 1 -5 < 1 We know 5 is greater than
-1 Multiply both sides by -1 Is -5 less than or greater than 1? You
know -5 is less than 1, so you should use < What happened to the
inequality symbol to keep the inequality statement true?
Slide 273
The direction of the inequality changes only if the number you
are using to multiply or divide by is negative. Helpful Hint
Slide 274
10234567 89 10 -2-3-4-5-6-7-8-9-10 Dividing each side by -3
changes the > to 18 -3y < 18 -3 -3 y < -6 Solve and graph.
A.
Slide 275
Divide each side by -7 Flip the sign because you divided by a
negative. 10234567 89 10 -2-3-4-5-6-7-8-9-10 -7m > -28 -7m <
-28 -7 -7 m < 4 Solve and graph. B.
Slide 276
Divide each side by 5. The sign does NOT change because you did
not divide by a negative. 5m > -25 5 5 m > -5 Solve and
graph. C. 10234567 89 10 -2-3-4-5-6-7-8-9-10
Slide 277
D. -8y > 32 Solve and graph. 10234567 89 10
-2-3-4-5-6-7-8-9-1010234567 89 10 -2-3-4-5-6-7-8-9-10 E. -9f >
-54
Slide 278
10234567 89 10 -2-3-4-5-6-7-8-9-10 You multiplied by a
negative. -r 2 < 5 -2 ( ) r > -10 Multiply both sides by the
reciprocal of -1/2. -r 2 > 5-2 () Why did the inequality
change?
