Algebraic inequalities by Vasile Cirtoaje

By Vasile Cirtoaje

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2 x 6 53. 264x8y3 3 54. 732 to evaluate the expression without using a calculator. 55. 23/2 56. 81/2 57. 93/4 58. 154 to evaluate the expression without using a calculator. 59. 103/2 60. 10003/2 61. 5 62. 0001)Ϫ1/3 14 1 PRELIMINARIES In Exercises 63–68, rationalize the denominator of the expression. 3 63. 2 1x 3 64. 1xy 2 66. 5x 13x 67. 1 3 2x 65. 68. 2y 98. 4x2 y 2z Ϫ 2x5y 2 ϩ 6x3y 2z2 99. 7a 4 Ϫ 42a2b2 ϩ 49a3b 13y 100. 3x 2/3 Ϫ 2x1/3 2x A y 101. eϪx Ϫ xeϪx 102. 2ye xy ϩ 2xy3e xy 2 In Exercises 69–74, rationalize the numerator of the expression.

Absolute Value Absolute Value The absolute value of a number a is denoted by ͉ a ͉ and is defined by | a| ϭ e a Ϫa if a Ն 0 if a Ͻ 0 22 1 PRELIMINARIES |a| |a| x –a 0 a FIGURE 4 Since Ϫa is a positive number when a is negative, it follows that the absolute value of a number is always nonnegative. For example, ͉ 5 ͉ ϭ 5 and ͉Ϫ5 ͉ ϭ Ϫ(Ϫ5) ϭ 5. Geometrically, ͉ a ͉ is the distance between the origin and the point on the number line that represents the number a (Figure 4). The absolute value of a number Absolute Value Properties If a and b are any real numbers, then Example Property 5 Property 6 | Ϫa | ϭ | a | | ab | ϭ | a | | b | Property 7 | a| a ` ` ϭ b | b| Property 8 | a ϩ b| Յ | a| ϩ | b| 1b 02 | Ϫ3 | ϭ Ϫ1Ϫ32 ϭ 3 ϭ | 3 | | 12 2 1Ϫ3 2 | ϭ | Ϫ6 | ϭ 6 ϭ 12 2 13 2 ϭ | 2 | | Ϫ3 | 1Ϫ3 2 | Ϫ3 | 3 3 ` ` ϭ ` ` ϭ ϭ 4 4 | Ϫ4 | 1Ϫ4 2 | 8 ϩ 1Ϫ5 2 | ϭ | 3 | ϭ 3 Յ | 8 | ϩ | Ϫ5 | ϭ 13 Property 8 is called the triangle inequality.

Find a relationship between x and y. y Solution By the definition of a circle, the distance between C(h, k) and P(x, y) is r. Using Formula (1), we have P(x, y) 21x Ϫ h2 2 ϩ 1y Ϫ k2 2 ϭ r r C(h, k) which, upon squaring both sides, gives the equation (x Ϫ h)2 ϩ (y Ϫ k)2 ϭ r 2 x that must be satisfied by the variables x and y. FIGURE 12 A circle with radius r and center C(h, k) A summary of the result obtained in Example 2 follows. Equation of a Circle An equation of the circle with center C(h, k) and radius r is given by (x Ϫ h)2 ϩ (y Ϫ k)2 ϭ r 2 (2) EXAMPLE 3 Find an equation of the circle with a.

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