CHAPTER 1
PROPERTIES:
1. Commutative: Order doesn't matter ­ works for addition and multiplication
a + b = b + a
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ab = ba
Subtraction rule:
*Subtracting a number is the same as adding its opposite a ­ b = a + ­b
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Division rule:
*Dividing by a number is the same as multiplying by its reciprocal a/b = a x 1/b
2. Associative: Grouping of numbers using parenthesis. Only works with addition and
multiplication
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Addition : (a + b) + c = a + ( b + c)
Multiplication: (ab)c=a(bc)
3. Identity Property of Addition:
a + 0 = a
Identity Property of Multiplication:
(0 is the additive identity)
a * 1 = a (1 is the multiplicative identity)
4. Inverse Property of Addition:
Inverse property of multiplication:
a + ­a = 0 (opposites are inverses in addition)
mult.)
a * 1/a = 1 (reciprocals are inverses in
5. Distribuitve property:
a( b + c) = ab + ac
Proof Example using properties:
Statements
Reasons
x + (5­x) = x + (5 ­x)
Given
x + (5­x) = x + ( 5 + ­x)
Subtraction Rule
x + (5 ­x) = x + ( ­x +5)
Commutative Property
x + (5 ­x) = (x + ­x ) + 5
Associative Property of addition
x + (5­x) = 0 + 5
Inverse Property of addition
x + (5­x) = 5
Identity Property
Unit Analysis:
• Take words out, write #s with units.
• Determine what final answer needs to be in. ex. Ft/sec, mile/hr.
• Use conversion factors to balance units.
• Cross out labels, keep #s.
• Flip fractions in order cross cancel labels.
• Multiply straight across.
50 mile​
5280 feet​
1 hour​
=​
50 x 5280​
= 73.3 ft/sec.
1 hour​
1 mile​
3600 sec​
3600
Simplify:
• Distribute
• Combine like terms
• Answer is an expression
• No equal sign
• Ex. 2 ( 3x – 1) + 4x – 15 à 10x – 17
Evaluate:
• Plug x into expression if given
• When using a negative number, you must put it in ( ).
• Ex. 2 ( 3x – 1) + 4x – 15 when x = ­2 à answer is ­37.
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Put numerator and denominators in their own sets of ( ).
Order of operations:
• Paranthesis
• Exponents
• Multiplication
• Division
• Addition
• Subtraction
Steps for solving equations:
• Simplify both sides of =.
• Get x terms to one side, #s to the other. ( adding or subtracting)
• Get x alone. ( multiply or divide)
• Check your answer in original equation.
Literal equations:
• Solve problems for specific variable
• Do opposite, to get new variable alone
• Ex. D = m/v solve for m. M = dv
*S = 2lw + 2wh + 2lh Solve for w.
Strategies for problem solving:
• Guess and check
• Drawing a diagram
• Make a table
• Unit analysis
• Write an equation
• Use formulas
*Look for a pattern
Literal equations:
• Solve problems for specific variable
• Do opposite, to get new variable alone
• Ex. D = m/v solve for m. M = dv
Inequalities:
• > greater than ( open circle)
• > greater than equal to (closed circle)
• < less than
• < less than equal to ( open circle)
• Solve just as if = sign ( closed circle)
• If multiplying or dividing by negative #, change the inequality symbol direction
• Shade were it makes it true ( put zero in for X)
* Compound Inequalities ­ AND / OR
Absolute Value:
• The distance the # is from 0
• Take the #s that fall in absolute value to see what makes the expression true – solve it for both
+ and – examples. Will get two answers.
• < and < stand for and, can be written as one ex. ­3 < y + 5 < 3 à shade inside.
• > and > stand for or, must be written as two separate. 7 – 2h > 9 or 7 – 2h < ­9. shade away.
* To solve equations. Rewrite with answer + and answer ­. If there are numbers outside of the
absolute value sign, move numbers to other side, before rewriting and solving.
* For inequalities, rewrite with same answer, but flip sign and change to negative.
*Extraneous solution­ Answers do not work ­ typically if there is a negative number after =