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Sect. 1.2 Operations & Properties of Real Numbers

Sect. 1.2 Operations & Properties of Real Numbers. Absolute Value Inequalities Addition, Subtraction, Opposites Multiplication, Division, Reciprocals Laws: Commutative, Associative, Distributive. Absolute Value – Concept. Positive distance from 0 on a number line

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Sect. 1.2 Operations & Properties of Real Numbers

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

  2. 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 1.2

  3. Absolute Value – Notation 1.2

  4. 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 1.2

  5. Addition of any Two Numbers 1.2

  6. Opposites (Additive Inverses) 1.2

  7. 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 1.2

  8. Multiplication & Division (+ or – ?) 1.2

  9. Fractions with One Negative Sign These are the Simplest Forms 1.2

  10. The Law of Reciprocals What 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 1.2

  11. Division by Zero 1.2

  12. Another look at Grouping Symbols and the Order of Operations 1.2

  13. 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 1.2

  14. 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 = 179(7)= 63 • Are Subtraction or Division Commutative ? 1.2

  15. Commutative Laws 1.2

  16. 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 ? 1.2

  17. The Associative Laws 1.2

  18. The Distributive Law and (a + b)c = ac + bc 1.2

  19. 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 1.2

  20. Let’s look at next time … • Section 1.3 Solving Equations • If we have time today, let’s review Reducing Numeric Fractions 1.2

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