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Chapter 2
Notes 2-1
Properties of Numbers
Vocabulary
Description
commutative property of addition
switching the numbers
commutative property of multiplication switching the numbers
Example
7+5=5+7
7x5=5x7
(com x)
associative property of addition
associative property of multiplication
grouping does not affect the answer
grouping does not affect the answer
(A + B ) + C = A + (B + C)
(A • B) C = A (B • C)
(assoc. of +)
additive identity
multiplicative identity
Notes 2-2
7+0=7
7•1=7
something added to get the same answer
something multiplied that is equal to one
Distributive Property
Vocabulary
distributive property
Description
giving something to each
passing one to each
Example
7 (2 + 3) =
(7 • 2) + (7 • 3)
Example from our homework
(5)
Notes 2-3
answer column
6 (23) =
(6 • 20) + (6 • 3) =
120 + 18 =
138
(5)138
Simplifying Algebraic Expressions
Vocabulary
Definition
Example
term
a number or product of a number and variable
3x + 55 + 4ab (3 terms)
constant
a term with no variable – just a number
55 (above)
like terms
terms with the same variable
coefficient
the number in front of a variable (multiply)
3g + 2b + 5g + 4b (add bs, gs)
simplify an expression replace the expression with an equivalent expression
(combining like terms)
using the fewest terms possible
(4) coefficients:3,5 like terms: none constants:-3
(answer)
3x + 4ab (3 first, 4 second)
3g + 2b + 5g + 4b =
8g + 6b
(4) 3,5 ; none ; -3
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Practice using NLVM – algebra – grades 6-8 “Algebra Balance Scales”
Notes 2-5
ONLINE FUN
Solving Equations by Adding or Subtracting
The opposite of subtraction is addition.
The opposite of addition
is subtraction.
Example
Vocabulary
inverse operation
solution
justify
Description
UNDO Do the opposite to both sides of the equal sign
answer - must be in [x = ____ ] form
tell which operation/property is used for each step
STEPS
1. Identify your variable.
2. “What are they doing to your variable?”
3. UNDO – Use the inverse operation
TO BOTH SIDES of the equal sign.
4. Solve.
x−4 = 8
+4 = +4
x = 12
Another Example
y – 7 = 12
+7 = +7
y
= 19
The variable is “y”.
They are subtracting 7.
ADD 7 to both sides of equation.
“subtract 7 and add 7” form a zero pair
Yes these are easy NOW. We are practicing the process, for bigger problems later. SHOW YOUR WORK IN STEPS.
Notes 2-6
Solving Equations by Multiplying or Dividing
The opposite of division is multiplication
The opposite of multiplication is division.
Vocabulary
inverse operation
solution
Example
Description
UNDO Do the opposite to both sides of the equal sign
answer - must be in [x= ____ ] form
STEPS
1. Identify your variable.
2. “What are they doing to your variable?”
Another Example
3. UNDO – Use the inverse operation
TO BOTH SIDES of the equal sign.
4. Solve.
•7 = •7
y = 2
7 ..
4x = 8
÷4 = ÷4
x = 2
The variable is “y”.
They are dividing by 7.
. .
MULTIPLY BY 7 on both sides.
..
y
= 14
“divide by 7 and multiply by 7” cancel out
Yes these are easy NOW. We are practicing the process, for bigger problems later. SHOW YOUR WORK IN STEPS.
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Notes 2-8
Inequalities and their Graphs
Vocabulary
inequality
solution
open dot
closed dot
shade
Description
a number sentence with a <, >, ≤, or ≥ sign
multiple answers
does not include (I’m not touching you)
includes ( AND this one)
make the line darker (stay ON the line)
Example
a > 90
an A is 96, 92, 91 . . .
A ≠ 90 (90 is an A-)
A ≥ 91 (91 too)
greater than
>
less than
<
greater than and equal to
≥
less than and equal to
≤
examples
x≥5
y > 90
closed dot because it does = 5 too
think of a number bigger than 5
the arrow goes toward THAT number
Notes 2-9
open dot because it does NOT include 90
think of a number greater than 90
the arrow goes toward THAT number
Solving Inequalities by Adding or Subtracting
The opposite of subtraction is addition.
The opposite of addition
is subtraction.
STEPS
Example
1. Identify your variable.
y – 7 ≤ 12
2. “What are they doing to your variable?”
..
3. UNDO – Use the inverse operation
+ 7 .. + 7
TO BOTH SIDES of the inequality sign.
..
4. Solve.
y ≤ 19
The variable is “y”.
They are subtracting 7
ADD 7 to both sides of the “river”.
“subtract 7 and add 7” form a zero pair
5. Graph on a number line
Notes 2-10
Solving Inequalities by Multiplying or Dividing
The opposite of division is multiplication
The opposite of multiplication is division.
STEPS
Example
1. Identify your variable.
y ≤ 2
2. “What are they doing to your variable?”
7 ..
Another Example
y ≤ 2
−7..
. .
3. UNDO – Use the inverse operation
• 7 ..
•7
• −7 ..
..
TO BOTH SIDES of the inequality sign.
4. Solve.
. .
y
≤ 14
• −7
..
y
≥
−14
When you multiply or divide by a NEGATIVE number, it switches the inequality
symbol.