SECTION A.6 Solving Equations A47 Check: ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) − − = ⋅ − − = ⋅ = − −=− ⋅−=−−= x x x x 2 1 1 2 2 1 2 1 3 1 3 5 2 5 2 5 2 2 5 3 1 3 Since the two expressions are equal, the solution checks. The solution set is { }2 . 2 Solve Rational Equations A rational equation is an equation that contains a rational expression. Examples of rational equations are + = − + − − = + x x x x x 3 1 2 1 7 and 5 4 3 2 To solve a rational equation, multiply both sides of the equation by the least common multiple of the denominators of the rational expressions that make up the rational equation. Need to Review? Least common multiples are discussed in Section A.5, pp. A39–A40. Solving a Rational Equation Solve the equation: ( )( ) − = − + − − x x x x 3 2 1 1 7 1 2 Solution EXAMPLE 3 First, note that the domain of the variable is { } ≠ ≠ x x x 1, 2 . Clear the equation of rational expressions by multiplying both sides by the least common multiple of the denominators of the three rational expressions, ( )( ) − − x x 1 2 . ( )( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) − = − + − − − − − = − − − + − − − = − − − + − − − − − = − + − = + = = x x x x x x x x x x x x x x x x x x x x x x x x x x 3 2 1 1 7 1 2 1 2 3 2 1 2 1 1 7 1 2 3 3 1 2 1 1 1 2 7 1 2 3 3 2 7 3 3 5 2 8 4 Multiply both sides by x x 1 2 . ( )( ) − − Divide out common factors on the left. Use the Distributive Property on both sides; divide out common factors on the right. Rewrite the equation. Combine like terms. Add 3 to both sides; subtract x from both sides. Divide by 2. Check: − = − = x 3 2 3 4 2 3 2 ( )( ) ( )( ) − + − − = − + − − = + ⋅ = + = = x x x 1 1 7 1 2 1 4 1 7 4 1 4 2 1 3 7 3 2 2 6 7 6 9 6 3 2 Since the two expressions are equal, = x 4 checks. The solution set is { }4 . Now Work problem 47 Quadratic Equations Quadratic equations are equations such as + + = − = − = x x x x x 2 8 0 3 5 0 9 0 2 2 2 A general definition is given next. Now Work problem 35
RkJQdWJsaXNoZXIy NjM5ODQ=