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![Page 1: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/1.jpg)
Tools of Algebra:
Variables and Expressions; Exponents and PEMDAS;
Working with Integers; Applying the Distributive Property; and Identifying
Properties of Real Numbers Compiled and adapted by
Lauren McCluskey
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Credits• “Algebra I” by Monica and Bob
Yuskaitis• “Interesting Integers” by Monica and
Bob Yuskaitis• “Multiplying Integers”• “Dividing Integers” • “Order of Operations” • “Properties” by D. Fisher• “Coordinate Plane” by Christine Berg• Prentice Hall Algebra I
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Algebra I
By Monica Yuskaitis
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Variable
• Variable – A variable is a letter or symbol that represents a number (unknown quantity).
• 8 + n = 12
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Expression
• Algebraic expression – a group of numbers, symbols, and variables that express an operation or a series of operations.
• m + 8• r – 3
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Evaluate
• Evaluate an algebraic expression – To find the value of an algebraic expression by substituting numbers for variables.
• m + 8 m = 2 2 + 8 = 10• r – 3 r = 5 5 – 3 = 2
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Simplify
Simplify – Combine like terms and complete all operations
• m + 8 + m 2 m + 8• (2m x 2) + 8n 4m + 8n
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Words That Lead to Addition
• Sum
• More than
• Increased
• Plus
• Altogether
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Words That Lead to Subtraction
• Decreased
• Less
• Difference
• Minus
• How many more
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Write Algebraic Expressionsfor These Word Phrases
• Ten more than a number
• A number decrease by 5
• 6 less than a number
• A number increased by 8
• The sum of a number & 9
• 4 more than a number
n + 10
w - 5
x - 6
n + 8n + 9
y + 4
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Types of Equations:
• Equations may be:
• ‘True’ (when the expressions on both sides of the equal sign are equivalent)
• ‘False’ (when the expressions on both sides of the equal sign are not equivalent)
• ‘Open sentences’ (when they contain one or more variables.
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Complete This Table
n 2n - 3 y5 *2 - 3
10 *2 - 3
21 *2 - 3
32 *2 - 3
173961
7
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ExponentsExponents influence only that which they touch directly. For example:
40 - d2 + cd * 3
(for c= 2 and d= 5)
40 - (5 * 5) + (2 * 5) * 3
40 - 25 + (10 * 3)
40 - 25 + 30
70 - 25
50
Now you try one: 40 + (-d)2 + cd *3
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Check your answer:
40 + (-5 * -5) + (5 * 2) * 340 + 25 + (10 * 3)40 + 25 + 30 95
*Note: In this case you multiply (-5) * (-5) because of the parentheses.
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Try one more:
40 - d2 + cd * 3
(for c = 2 and d= -5)
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Check your answer:
40 - (-5)(-5) + (-5) (2) * 3
40 - (25) + [(-10) * 3]
40 + (-25) + (-30)
40 + (-55)
-15
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Exploring Real Numbers:
• Natural Numbers may also be known as counting numbers {1, 2, 3…}
• Whole Numbers include zero {0, 1, 2, …}
• Integers include negative numbers
{…-2, -1, 0, 1, 2, …}
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Exploring Real Numbers
• Rational Numbers can be written as either terminating or repeating decimals
• Irrational Numbers do not terminate or repeat when written as decimals
• Real Numbers include all rational and irrational numbers
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Order of Operations
A standard way to simplify mathematical
expressions and equations.
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Purpose• Avoids Confusion• Gives Consistency
For example:8 + 3 * 4 = 11 * 4 = 44
Or does it equal8 + 3 * 4 = 8 + 12 = 20
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Order of operations are a set of rules that mathematicians have agreed to follow to avoid mass CONFUSION when simplifying mathematical expressions or equations. Without these simple, but important rules, learning mathematics would
be maddening.
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The Rules
1) Simplify within Grouping Symbols
( ), { }, [ ], | |2) Simplify Exponents
Raise to Powers3) Complete Multiplication and
Division from Left to Right4) Complete Addition and
Subtraction from Left to Right
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Back to Our Example
For example:
8 + 3 * 4 = 11 * 4 = 44
Or does it equal
8 + 3 * 4 = 8 + 12 = 20Using order of operations, we do the multiplication first. So what’s ouranswer? 20
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How can we remember it?
• Parenthesis - Please
• Exponents - Excuse
• Multiplication - My
• Division - Dear
• Addition - Aunt
• Subtraction - Sally
OR: ‘PEMDAS’
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Adding / Subtracting Real Numbers
• Inverse property:
“For every real number n, there is an additive inverse -n such that
n + (-n) = 0.
We use the inverse property to solve equations.
from Prentice hall Algebra I
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Matrices
by Lauren McCluskey
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Adding Matrices
[ ] [ ] -5 + (-3) 2.7 + (-3.9) 7 + (-4) -3 + 2
[ ]
-5 2.77 -3
-3 -3.9-4 2
-8 -1.23 -1
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Try It!
[ ] + [ ]
[ ]
-5 -3/4
1/2 -1-4 7/8
3/4 0
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Check your answer:
-4 + (-5) 7/8 + (-3/4)3/4 + 1/2 0 + (-1)
[ ]-9 1/8
1 1/4 -1
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Scalar Multiplication (Matrices)
[ ] (-0.1)(-47) (-0.1)(13) (-0.1)(-7.9)
(-0.1)(0.2) (-0.1)(-64) (-0.1)(0)
[ ]
-47-0.1 13 -7.90.2 -64 0
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Check your answer:
[ ]4.7 -1.3 0.79-0.02 6.4 0
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Try It!
[ ]
3/5 * ____ 3/5* _____
3/5 * ____ 3/5* _____
-253/5 3510/9 -15
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Check your answer:
3/5 * -25 = -15 3/5 * 35= 21
3/5 * 10/9= 2/3 3/5 * -15= -9
[ ]
-15 212/3 -9
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For more practice:
• Go to pages 27-30 for +;
or pages 43 and 45 for * (scalar).
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Interesting Integers!
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What You Will Learn• Some definitions related to
integers.
• Rules for adding and subtracting integers.
• A method for proving that a rule is true.
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Definition
• Positive number – a number greater than zero.
0 1 2 3 4 5 6
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Definition
• Negative number – a number less than zero.
0 1 2 3 4 5 6-1-2-3-4-5-6
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Definition
• Opposite Numbers – numbers that are the same distance from zero in the opposite direction
0 1 2 3 4 5 6-1-2-3-4-5-6
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Definition
• Integers – Integers are all the whole numbers and all of their opposites on the negative number line including zero.
7 opposite -7
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Definition• Absolute Value – The size of a
number with or without the negative sign.
The absolute value of 9 or of –9 is 9.
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Negative Numbers Are Used to Measure Temperature
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Negative Numbers Are Used to Measure Under Sea Level
0102030
-10-20-30-40-50
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Negative Numbers Are Used to Show Debt
Let’s say your parents bought a car buthad to get a loan from the bank for $5,000.When counting all their money they add in -$5.000 to show they still owe the bank.
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Integer Addition Rules• Rule #1 – If the signs are the same,
pretend the signs aren’t there. Add the numbers and then put the sign of the addends in front of your answer.
OR:
• Think Teams: Which team won? How much did they win by?
9 + 5 = 14 -9 + -5 = -14
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Solve the Problems• -3 + -5 =
• (+3) + (+4) =
• -6 + -7 =
• -9 + -9 =
-8
-18
-13
7
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Integer Addition Rules• Rule #2 – If the signs are different pretend the
signs aren’t there. Subtract the smaller from the larger one and put the sign of the one with the larger absolute value in front of your answer.
OR• Think Teams: Which team won? How much
did they win by?
-9 + +5 =9 - 5 = 4
Larger abs. value
Answer = - 4
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Solve These Problems
• 3 + -5 =• -4 + 7 =• (+3) + (-4) =• -6 + 7 = • 5 + -9 =• -9 + 9 =
-25 – 3 = 2
0 -4
1-1
3
9 – 9 = 0
9 – 5 = 4
7 – 6 = 14 – 3 = 1
7 – 4 = 3
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One Way to Add Integers Is With a Number Line
0 1 2 3 4 5 6-1-2-3-4-5-6
When the number is positive, countto the right.
When the number is negative, countto the left.
+-
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Adding on a Number Line
0 1 2 3 4 5 6-1-2-3-4-5-6+
-
+3 + -5 = -2
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Adding Integers Is With a Number Line
0 1 2 3 4 5 6-1-2-3-4-5-6+
-
+6 + -4 = +2
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Adding Integers Is With a Number Line
0 1 2 3 4 5 6-1-2-3-4-5-6+
-
+3 + -7 = -4
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Integer Subtraction RuleSubtracting a negative number is the same as adding its opposite. Change the signs and add.
2 – (-7) is the same as
2 + (+7)
2 + 7 = 9!
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Integer Subtraction RuleSubtracting a negative number is the same as adding its opposite. Change the signs and add.
2 – (-7) is the same as
2 + (+7)
2 + 7 = 9!
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Here are some more examples.
12 – (-8)
12 + (+8)
12 + 8 = 20
-3 – (-11)
-3 + (+11)
-3 + 11 = 8
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Check Your Answers
1. 8 – (-12) = 8 + 12 = 20
2. 22 – (-30) = 22 + 30 = 52
3. – 17 – (-3) = -17 + 3 = -14
4. –52 – 5 = -52 + (-5) = -57
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Multiplying / Dividing Real Numbers
Multiplying and dividing positive and negative numbers is easy when you remember the rules:
positive * positive = positive [+*+ = +] negative * negative = negative [- * - = -]positive * negative = negative [+ * - = - ]
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MULTIPLYING INTEGERS
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Problem 1
(-3)( 5)=
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Problem 2
(5)(-3) =
![Page 61: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/61.jpg)
Problem 3
(-2)(-10) =
![Page 62: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/62.jpg)
Problem 4
(-3)(8) =
![Page 63: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/63.jpg)
Problem 5
(-6)(8) =
![Page 64: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/64.jpg)
Problem 6
(3)(-9)=
![Page 65: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/65.jpg)
Problem 7
(-7)(-3) =
![Page 66: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/66.jpg)
Problem 8
(-9)(0) =
![Page 67: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/67.jpg)
Problem 9
(-9)(-7) =
![Page 68: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/68.jpg)
Problem 10
(16)(-10) =
![Page 69: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/69.jpg)
Problem 11
(9)(-5) =
![Page 70: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/70.jpg)
Problem 12
(-4)(-9) =
![Page 71: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/71.jpg)
Problem 13
(5)(-1) =
![Page 72: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/72.jpg)
Problem 14
(10)(-4) =
![Page 73: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/73.jpg)
Problem 15
(15)(-2) =
![Page 74: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/74.jpg)
Problem 16
(-5)(-11) =
![Page 75: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/75.jpg)
Problem 17
(-10)(4) =
![Page 76: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/76.jpg)
Problem 18
(-4)(7) =
![Page 77: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/77.jpg)
Problem 19
(-12)(5) =
![Page 78: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/78.jpg)
Problem 20
(-8)(-4) =
![Page 79: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/79.jpg)
Check your answers:1)-15 11) -452) -15 12) +363) +20 13) -5 4) -24 14) -405) -48 15) -30 6) -27 16) +557) +21 17) -408) 0 18) -289) +63 19) -6010) -160 20) +32
![Page 80: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/80.jpg)
DIVIDING INTEGERS
1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE
2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE
Remember:
![Page 81: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/81.jpg)
Problem 1
(-6)÷( 3)=
![Page 82: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/82.jpg)
Problem 2
(6) ÷(-3) =
![Page 83: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/83.jpg)
Problem 3
(-10) ÷ (-2) =
![Page 84: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/84.jpg)
Problem 4
(-16) ÷ (2) =
![Page 85: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/85.jpg)
Problem 5
(-12) ÷ (6) =
![Page 86: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/86.jpg)
Problem 6
(9) ÷ (-3)=
![Page 87: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/87.jpg)
Problem 7
(-8) ÷ (-4) =
![Page 88: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/88.jpg)
Problem 8
(-9) ÷ (0) =
![Page 89: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/89.jpg)
Problem 9
(-21) ÷ (-7) =
![Page 90: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/90.jpg)
Problem 10
(16) ÷ (-8) =
![Page 91: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/91.jpg)
Problem 11
(10) ÷ (-5) =
![Page 92: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/92.jpg)
Problem 12
(-12) ÷ (-6) =
![Page 93: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/93.jpg)
Problem 13
(5) ÷ (-1) =
![Page 94: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/94.jpg)
Problem 14
(10) ÷ (-5) =
![Page 95: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/95.jpg)
Problem 15
(15) ÷ (-3) =
![Page 96: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/96.jpg)
Problem 16
(-22) ÷ (-11) =
![Page 97: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/97.jpg)
Problem 17
(-16) ÷ (4) =
![Page 98: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/98.jpg)
Problem 18
(-14) ÷ (7) =
![Page 99: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/99.jpg)
Problem 19
(-12) ÷ (6) =
![Page 100: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/100.jpg)
Problem 20
(-8) ÷ (-4) =
![Page 101: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/101.jpg)
Check your answers: 1)-2 11) -22) -2 12) +23) +5 13) -54) -8 14) -25) -2 15) -56) -3 16) +27) +2 17) -48) 0 18) -29) +3 19) -210) -2 20) +2
![Page 102: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/102.jpg)
The Distributive Property
•You can use the Distributive Property
to multiply a sum or difference
by a number.
•You can also use the Distributive Property
to simplify algebraic expressions by
removing the parentheses.
![Page 103: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/103.jpg)
A good way to remember how to apply the Distributive Property is to visualize 2 rectangles:
3*1= 3
3 * x = 3x3
1x
So 3( x + 1) =(3 * x) + (3* 1)=3x + 3
![Page 104: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/104.jpg)
Try It!
a) 2/3(6y + 9)b) 0.25(6q + 32)c) (8- 3r) 5/16d) -4.5(b- 3)
![Page 105: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/105.jpg)
Check your answers:
a) 4y +6
b) 1.5q + 8
c) 2 1/2 + 15/16 r
d) -4.5b + 13.5
![Page 106: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/106.jpg)
by D. Fisher
![Page 107: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/107.jpg)
(2 + 1) + 4 = 2 + (1 + 4)Associative Associative Property of Property of AdditionAddition
![Page 108: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/108.jpg)
3 + 7 = 7 + 3
Commutative Commutative Property of Property of AdditionAddition
![Page 109: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/109.jpg)
8 + 0 = 8
Identity Identity Property of Property of AdditionAddition
..
![Page 110: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/110.jpg)
6 • 4 = 4 • 6
Commutative Commutative Property of Property of
MultiplicationMultiplication
![Page 111: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/111.jpg)
2(5) = 5(2)
Commutative Commutative Property of Property of
MultiplicationMultiplication
![Page 112: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/112.jpg)
3(2 + 5) = 3•2 + 3•5Distributive Distributive
PropertyProperty
![Page 113: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/113.jpg)
6(7•8) = (6•7)8
Associative Associative Property of Property of
MultiplicationMultiplication
![Page 114: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/114.jpg)
![Page 115: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/115.jpg)
6(3 – 2n) = 18 – 12n
Distributive Distributive PropertyProperty
![Page 116: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/116.jpg)
2x + 3 = 3 + 2x
Commutative Commutative Property of Property of AdditionAddition
![Page 117: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/117.jpg)
ab = ba
Commutative Commutative Property of Property of
MultiplicationMultiplication
![Page 118: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/118.jpg)
a + 0 = a
Identity Identity Property of Property of AdditionAddition
![Page 119: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/119.jpg)
a(bc) = (ab)c
Associative Associative Property of Property of
MultiplicationMultiplication
![Page 120: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/120.jpg)
a•1 = a
Identity Identity Property of Property of
MultiplicationMultiplication
![Page 121: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/121.jpg)
a +b = b + a
Commutative Commutative Property of Property of AdditionAddition
![Page 122: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/122.jpg)
a(b + c) = ab + ac
Distributive Distributive PropertyProperty
![Page 123: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/123.jpg)
a + (b + c) = (a +b) + c
Associative Associative Property of Property of AdditionAddition
![Page 124: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/124.jpg)
The Coordinate
PlaneBy: Christine BergEdited By:VTHamilton
![Page 125: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/125.jpg)
DefinitionThe plane formed when 2
perpendicular number lines intersect at their
zero points
Coordinate Plane
![Page 126: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/126.jpg)
Coordinate Plane
The perpendicular number lines form a
grid on the plane
![Page 127: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/127.jpg)
![Page 128: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/128.jpg)
X-axis
The horizontal number line
•Positive to the right
•Negative to the left
![Page 129: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/129.jpg)
Y-axis
The vertical
number line
•Positive upward
•Negative downward
![Page 130: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/130.jpg)
Origin
Where the x and y axes intersect at their zero
points
![Page 131: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/131.jpg)
![Page 132: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/132.jpg)
Quadrants
The x and y axes divide the coordinate plane into 4 parts called
quadrants
![Page 133: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/133.jpg)
III
III IV
![Page 134: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/134.jpg)
Ordered Pair
A pair of numbers (x , y) assigned to a point
on the coordinate plane
![Page 135: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/135.jpg)
Ordered Pair
(x , y)
X-coordinateX-coordinate
Y-coordinateY-coordinate
![Page 136: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/136.jpg)
Plotting a Point
Step 1:
Begin at the Origin
![Page 137: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/137.jpg)
Plotting a Point
Step 2:
Locate x on the x-axis
![Page 138: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/138.jpg)
Plotting a Point
Step 3:
Move up or down to the value of y
![Page 139: Tools of Algebra : Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties.](https://reader038.fdocuments.in/reader038/viewer/2022110101/56649eb05503460f94bb6403/html5/thumbnails/139.jpg)
Plotting a Point
Step 4:
Draw a dot and label the point