155 1.6 OtherTypes of Equations and Applications (b) x x - 2 = 2 x - 2 + 2 1x - 22a x x - 2b = 1x - 22a 2 x - 2b + 1x - 222 Multiply by the LCD, x - 2, where x ≠2. x = 2 + 1x - 222 Divide out common factors. x = 2 + 2x - 4 Distributive property -x = -2 Solve the linear equation. x = 2 Proposed solution The proposed solution is 2. However, the variable is restricted to real numbers except 2. If x = 2, then not only does it cause a zero denominator, but also multiplying by x - 2 in the first step is multiplying both sides by 0, which is not valid. Thus, the solution set is ∅. S Now Try Exercises 17 and 19. EXAMPLE 2 Solving Rational Equations That Lead to Quadratic Equations Solve each equation. (a) 3x + 2 x - 2 + 1 x = -2 x2 - 2x (b) -4x x - 1 + 4 x + 1 = -8 x2 - 1 SOLUTION (a) 3x + 2 x - 2 + 1 x = -2 x2 - 2x 3x + 2 x - 2 + 1 x = -2 x1x - 22 Factor the last denominator. x1x - 22a 3x + 2 x - 2 b + x1x - 22a 1 xb = x1x - 22a -2 x1x - 22 b Multiply by x1x - 22, x ≠0, 2. x13x + 22 + 1x - 22 = -2 Divide out common factors. 3x2 + 2x + x - 2 = -2 Distributive property 3x2 + 3x = 0 Write in standard form. 3x1x + 12 = 0 Factor. 3x = 0 or x + 1 = 0 Zero-factor property x = 0 or x = -1 Proposed solutions Because of the restriction x ≠0, the only valid proposed solution is -1. CHECK 31-12 + 2 -1 - 2 + 1 -1 ≟ -2 1-122 - 21-12 Let x = -1 in the original equation. -1 -3 - 1≟ -2 3 Simplify each fraction. - 2 3 = - 2 3 ✓ True The solution set is 5-16. Set each factor equal to 0.
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