192 CHAPTER 1 Equations and Inequalities Concepts Examples Solve. -31x + 42 + 2x 66 -3x - 12 + 2x 66 -x 618 x 7 -18 Multiply by -1. Change 6 to 7. The solution set is 1-18, ∞2. x2 + 6x … 7 x2 + 6x - 7 = 0 Corresponding equation 1x + 721x - 12 = 0 Factor. x + 7 = 0 or x - 1 = 0 Zero-factor property x = -7 or x = 1 Solve each equation. The intervals formed are 1-∞, -72, 1-7, 12, and 11, ∞2. Test values show that values in the intervals 1-∞, -72 and 11, ∞2 do not satisfy the original inequality, while those in 1-7, 12 do. Because the symbol … includes equality, the endpoints are included. The solution set is 3-7, 14. x x + 3 Ú 5 x + 3 x x + 3 - 5 x + 3 Ú 0 x - 5 x + 3 Ú 0 The values -3 and 5 make either the numerator or the denominator 0. The intervals formed are 1-∞, -32, 1-3, 52, and 15, ∞2. The value -3 must be excluded and 5 must be included. Test values show that values in the intervals 1-∞, -32 and 15, ∞2 yield true statements. The solution set is 1-∞, -32 ´35, ∞2. Solve. 1x + 122/3 + 1x + 121/3 - 6 = 0 u2 + u - 6 = 0 Let u = 1x + 121/3. 1u + 321u - 22 = 0 u + 3 = 0 or u - 2 = 0 u = -3 or u = 2 1x + 121/3 = -3 or 1x + 121/3 = 2 x + 1 = -27 or x + 1 = 8 Cube. x = -28 or x = 7 Subtract 1. Both solutions check. The solution set is 5-28, 76. 1.6 Other Types of Equations and Applications Power Property If P and Q are algebraic expressions, then every solution of the equation P = Q is also a solution of the equation Pn = Qn, for any positive integer n. Quadratic in Form An equation in the form au2 +bu +c =0, where a≠0 and u is an algebraic expression, can be solved using a substitution variable. If the power property is applied, or if both sides of an equation are multiplied by a variable expression, check all proposed solutions in the original equation. 1.7 Inequalities Properties of Inequality Let a, b, and c represent real numbers. 1. If a *b, then a +c *b +c. 2. If a *b and if c +0, then ac *bc. 3. If a *b and if c *0, then ac +bc. Solving a Quadratic Inequality Step 1 Solve the corresponding quadratic equation. Step 2 Identify the intervals determined by the solutions of the equation. Step 3 Use a test value from each interval to determine which intervals form the solution set. Solving a Rational Inequality Step 1 Rewrite the inequality, if necessary, so that 0 is on one side and there is a single fraction on the other side. Step 2 Determine the values that will cause either the numerator or the denominator of the rational expression to equal 0. These values determine the intervals on the number line to consider. Step 3 Use a test value from each interval to determine which intervals form the solution set.
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