sect. 1.2 operations & properties of real numbers

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Sect. 1.2Operations & Properties of Real Numbers Absolute Value Inequalities Addition, Subtraction, Opposites Multiplication, Division, Reciprocals Laws: Commutative, Associative, Distributive

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Absolute Value – Concept Positive distance from 0 on a number line

How far is the number 6 from the 0 point? 6 units

How far is -4.75 from the 0 point? 4.75 units

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Absolute Value – Notation

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Inequalities – Concept Use the number line to understand

< less than > more than

Express the relation of -4 to -7 Either -7 < -4 or -4 > -7 both ways are correct

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Addition of any Two Numbers

14311 64)13(51

02323 8146

13)37(24 2.6)2.6(0

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Opposites (Additive Inverses)

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Opposites and Absolute Value Opposites are a pair of numbers that sum to 0

11.2 and -11.2 -3/17 and 3/17 0 and 0 (zero is it’s own opposite)

The Absolute Value of a number is the positive value of it’s pair of opposites |11.2| = 11.2 and |-11.2| = 11.2 |-3/17| = 3/17 and |3/17| = 3/17

Absolute value brackets can hold expressions |3 – 5| = |-2| = 2 |5 – 3| = |2| = 2 -|22| = -22 and -|-22| = -22

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Multiplication & Division (+ or – ?)

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Fractions with One Negative Sign

ba

ba

ba

ba

Similarly

:These are the Simplest Forms

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The Law of ReciprocalsWhat is 4’s reciprocal?

¼ because 4 (¼) = 1

What is 3½ ’s reciprocal?

2/7 Because (7/2)(2/7) = 1

a and 1/a are Multiplicative Inverses

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Division by Zero

03x

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Another look at Grouping Symbols and the Order of Operations

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Equivalent Expressions Two expressions are Equivalent when they have the

same values for all possible replacements Are the following two expressions equivalent?

3x + 4 and x – 3 + 2x + 7 Yes –

when simplified, the 2nd expression matches the 1st

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I Must Remember: Commutative An operation is Commutative when

Two values can switch positions, and the result is the same.

COMMUTATIVE

Addition and Multiplication are Commutative 6 + 11 = 17 7(9) = 63 11 + 6 = 17 9(7) = 63

Are Subtraction or Division Commutative ?

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Commutative Laws

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I Must Remember: Associative Three values can be computed in different order and

the result is the same. (same operations)

ASSOCIATIVE ASSOCIATIVE

Addition and Multiplication are Associative (19 + 4) + 6 = 23 + 6 = 29 (3)(7)(5) = 21(5) = 105 19 + (4 + 6) = 19 + 10 = 29 (3)(7)(5) = 3(35) = 105

Are Subtraction or Division Associative ?

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The Associative Laws

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The Distributive Law

301218

)4(3)6(3)46(3

1510)15(10

)3(5)2(5)32(5

xxxx

and (a + b)c = ac + bc

yy

yyyyy

23

)1()3()13(

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Let’s play Name That Law! x + 5 + y = x + y + 5

Commutative … COM 3a + 6 = 3(a + 2)

Distributive … DIST 7x(1 / x) = 7

Reciprocals … RECIP (x + 5) + y = x + (5 + y)

Associative … ASSOC 4(a + 2b) = 8b + 4a

COM, then DIST or DIST, then COM

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Let’s look at next time … Section 1.3 Solving Equations

If we have time today, let’s reviewReducing Numeric Fractions