154 CHAPTER 1 Equations and Inequalities Rational Equations A rational equation is an equation that has a rational expression for one or more terms. To solve a rational equation, multiply each side by the least common denominator (LCD) of the terms of the equation to eliminate fractions, and then solve the resulting equation. A value of the variable that appears to be a solution after each side of a rational equation is multiplied by a variable expression (the LCD) is called a proposed solution. Because a rational expression is not defined when its denominator is 0, proposed solutions for which any denominator equals 0 are excluded from the solution set. Be sure to check all proposed solutions in the original equation. 1.6 Other Types of Equations and Applications ■ Rational Equations ■ Work Rate Problems ■ Equations with Radicals ■ Equations with Rational Exponents ■ Equations Quadratic in Form EXAMPLE 1 Solving Rational Equations That Lead to Linear Equations Solve each equation. (a) 3x - 1 3 - 2x x - 1 = x (b) x x - 2 = 2 x - 2 + 2 SOLUTION (a) The least common denominator is 31x - 12, which is equal to 0 if x = 1. Therefore, 1 cannot possibly be a solution of this equation. 3x - 1 3 - 2x x - 1 = x 3 1x - 12a 3x - 1 3 b - 31x - 12a 2x x - 1b = 31x - 12x Multiply by the LCD, 31x - 12, where x ≠1. 1x - 1213x - 12 - 312x2 = 31x - 12x Divide out common factors. 3x2 - 4x + 1 - 6x = 3x2 - 3x Multiply. 1 - 10x = -3x Subtract 3x2. Combine like terms. 1 = 7x Solve the linear equation. x = 1 7 Proposed solution The proposed solution 1 7 meets the requirement that x ≠1 and does not cause any denominator to equal 0. Substitute to check for correct algebra. CHECK 3x - 1 3 - 2x x - 1 = x Original equation 3A1 7B - 1 3 - 2A1 7B 1 7 - 1 ≟1 7 Let x = 1 7 . - 4 21 - a- 1 3b ≟1 7 Simplify the complex fractions. 1 7 = 1 7 ✓ True The solution set is E1 7F.
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