By James Stewart, Lothar Redlin, Saleem Watson
This top promoting writer staff explains techniques easily and obviously, with out glossing over tough issues. challenge fixing and mathematical modeling are brought early and strengthened all through, delivering scholars with a fantastic starting place within the ideas of mathematical pondering. complete and calmly paced, the publication offers whole insurance of the functionality suggestion, and integrates an important volume of graphing calculator fabric to aid scholars improve perception into mathematical rules. The authors' cognizance to element and readability, just like present in James Stewart's market-leading Calculus publication, is what makes this publication the industry chief.
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Additional resources for Algebra and Trigonometry , Third Edition
Therefore (1, 3) 0 1 3 11, 32 ʜ 32, 74 ϭ 5x 0 1 Ͻ x Ͻ 3 or 2 Յ x Յ 76 [2, 7] 0 2 ϭ 5x 0 1 Ͻ x Յ 76 ϭ 11, 74 7 This set is illustrated in Figure 8. (1, 7] 0 1 F I G U R E 8 11, 32 ʜ 32, 74 | _3 |=3 _3 ■ NOW TRY EXERCISE 47 7 ▼ Absolute Value and Distance | 5 |=5 0 5 FIGURE 9 The absolute value of a number a, denoted by 0 a 0 , is the distance from a to 0 on the real number line (see Figure 9). Distance is always positive or zero, so we have 0 a 0 Ն 0 for every number a. Remembering that Ϫa is positive when a is negative, we have the following definition.
1Ϫa2 1Ϫb2 ϭ ab 1Ϫ42 1Ϫ32 ϭ 4 # 3 3. 1Ϫa2b ϭ a1Ϫb2 ϭ Ϫ1ab2 1Ϫ527 ϭ 51Ϫ72 ϭ Ϫ15 # 72 5. Ϫ1a ϩ b2 ϭ Ϫa Ϫ b Ϫ13 ϩ 52 ϭ Ϫ3 Ϫ 5 6. Ϫ1a Ϫ b2 ϭ b Ϫ a Ϫ15 Ϫ 82 ϭ 8 Ϫ 5 Property 6 states the intuitive fact that a Ϫ b and b Ϫ a are negatives of each other. Property 5 is often used with more than two terms: Ϫ1a ϩ b ϩ c2 ϭ Ϫa Ϫ b Ϫ c EXAMPLE 4 Using Properties of Negatives Let x, y, and z be real numbers. (a) Ϫ13 ϩ 22 ϭ Ϫ3 Ϫ 2 (b) Ϫ1x ϩ 22 ϭ Ϫx Ϫ 2 (c) Ϫ1x ϩ y Ϫ z 2 ϭ Ϫx Ϫ y Ϫ 1Ϫz2 ϭ Ϫx Ϫ y ϩ z NOW TRY EXERCISE 9 Property 5: –(a + b) = –a – b Property 5: –(a + b) = –a – b Property 5: –(a + b) = –a – b Property 2: –(–a) = a ■ ▼ Multiplication and Division The number 1 is special for multiplication; it is called the multiplicative identity because a и 1 ϭ a for any real number a.
2 (b) The union of two intervals consists of the numbers that are in either one interval or the other (or both). Therefore (1, 3) 0 1 3 11, 32 ʜ 32, 74 ϭ 5x 0 1 Ͻ x Ͻ 3 or 2 Յ x Յ 76 [2, 7] 0 2 ϭ 5x 0 1 Ͻ x Յ 76 ϭ 11, 74 7 This set is illustrated in Figure 8. (1, 7] 0 1 F I G U R E 8 11, 32 ʜ 32, 74 | _3 |=3 _3 ■ NOW TRY EXERCISE 47 7 ▼ Absolute Value and Distance | 5 |=5 0 5 FIGURE 9 The absolute value of a number a, denoted by 0 a 0 , is the distance from a to 0 on the real number line (see Figure 9).