By M Harris, David Mower
Possibilities is a brand new five-level direction for teens. Modules of topic-based devices offer wealthy, modern content material in line with a large choice of data issues. With a discovery method of grammar and an in advance specialise in vocabulary, possibilities guarantees the best language studying for college students.
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Extra info for Opportunities: Beginner Teacher's Book
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2yz2 2 2 48. 18 a2z2 A 12 a3z4 B 3 2 2 1u3√Ϫ2 2 3 3a Ϫ1 b b3 25 1rs2 2 3 66. 69. a ✎ 73. ✎ 75. Integer Exponents 89–90 ■ Write the number indicated in each statement in scientific notation. 28. 3Ϫ1 Ϫ 3Ϫ3 8 2 35. ✎ 15. 2 10 104 A 131 B 0 A 23 B 6 A 49 B Ϫ3 29–46 12. Ϫ6 0 2Ϫ3 20. 0 3 A 53 B 0 2Ϫ1 22. A 23 B Ϫ3 25. 3 2 7 16. 54 # 5Ϫ2 ✎ 19. 9. 12 2 # 22 ✎ 67. 05 ϫ 104 2 12 5xyϪ2 97. Prove the given Laws of Exponents for the case in which m and n are positive integers and m Ͼ n. (a) Law 2 (b) Law 5 xϪ1yϪ3 98.
Combining Rational Numbers with Irrational Numbers Is 12 ϩ 22 rational or irrational? 3 The Real Number Line and Order LEARNING OBJECTIVES After completing this section, you will be able to: ■ ■ ■ ■ ■ Graph numbers on the real line Use the order symbols Ͻ, Յ, Ͼ, Ն Work with set and interval notation Find and use absolute values of real numbers Find distances on the real line ■ The Real Line The real numbers can be represented by points on a line, as shown in Figure 1. The positive direction (toward the right) is indicated by an arrow.
The notation 1a, q2 stands for the set of all numbers that are greater than a, so the symbol q simply indicates that the interval extends indefinitely far in the positive direction. The following table lists the nine possible types of intervals. When these intervals are discussed, we will always assume that a Ͻ b. Notation 1a, b2 5x Ȋ a Ͻ x Ͻ b6 3a, b2 3a, b4 1a, b4 1a, q2 3a, q2 1Ϫq, b2 1Ϫq, b4 1Ϫq, q2 Ǡ Set description E X AM P L E 3 | Graph 5x Ȋ a Յ x Յ b6 a b 5x Ȋ a Յ x Ͻ b6 a b 5x Ȋ a Ͻ x Յ b6 a b 5x Ȋ a Ͻ x6 a b 5x Ȋ a Յ x6 a 5x Ȋ x Ͻ b} a b 5x Ȋ x Յ b6 b ( ޒset of all real numbers) Graphing Intervals Express each interval in terms of inequalities, and then graph the interval.