25 R.3 Real Number Operations and Properties Examples of Properties 1–4: 0 -150 = 15 and 15 Ú 0. Property 1 0 -100 = 10 and 0 100 = 10, so 0 -100 = 0 100 . Property 2 0 50 # 0 -4 0 = 5 # 4 = 20 and 0 51-42 0 = 0 -200 = 20, so 0 50 # 0 -4 0 = 0 51-42 0 . Property 3 0 20 0 30 = 2 3 and ` 2 3 ` = 2 3 , so 0 20 0 30 = ` 2 3 ` . Property 4 Example of the triangle inequality: 0 a + b0 = 0 3 + 1-72 0 = 0 -4 0 = 4 Let a = 3 and b = -7. 0 a0 + 0 b0 = 0 30 + 0 -7 0 = 3 + 7 = 10 Thus, 0 a + b0 … 0 a0 + 0 b0 . Property 5 LOOKING AHEAD TO CALCULUS One of the most important definitions in calculus, that of the limit, uses absolute value. The symbols P (epsilon) and d (delta) are often used to represent small quantities in mathematics. Suppose that a function ƒ is defined at every number in an open interval I containing a, except perhaps at a itself. Then the limit of ƒ1x2 as x approaches a is L, written lim xSa ƒ1x2 = L, if for every P 70 there exists a d 70 such that 0 ƒ1x2 - L0 6 P whenever 0 6 0 x - a 0 6d. Properties of Absolute Value Let a and b represent real numbers. Property Description 1. 0 a0 #0 The absolute value of a real number is positive or 0. 2. 0 −a0 = 0 a0 The absolute values of a real number and its opposite are equal. 3. 0 a0 # 0 b0 = 0 ab0 The product of the absolute values of two real numbers equals the absolute value of their product. 4. 0 a0 0 b0 = ` a b ` 1 b 302 The quotient of the absolute values of two real numbers equals the absolute value of their quotient. 5. 0 a +b0 " 0 a0 + 0 b0 (triangle inequality) The absolute value of the sum of two real numbers is less than or equal to the sum of their absolute values. SOLUTION (a) ` - 5 8 ` = 5 8 (b) -0 80 = -182 = -8 (c) -0 -8 0 = -182 = -8 (d) 0 2p0 = 2p S Now Try Exercises 11 and 15. EXAMPLE 1 Evaluating Absolute Values Evaluate each expression. (a) ` - 5 8 ` (b) -0 80 (c) -0 -8 0 (d) 0 2x 0 , for x = p Consider the following properties of absolute value.
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