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Section 6.5 / Solving Equations Containing Fractions 6.5 Objective A 315 Solving Equations Containing Fractions To solve an equation containing fractions Recall that to solve an equation containing fractions, clear denominators by multiplying each side of the equation by the LCM of the denominators. Then solve for the variable. Solve: HOW TO 3x 1 2 7 4 3 6 3x 1 2 7 4 3 6 3x 1 2 7 12 12 4 3 6 3x 1 2 7 12 12 12 4 3 6 3 12 3x 1 1 4 1 4 • The LCM is 12. To clear denominators, multiply each side of the equation by the LCM. • Simplify using the Distributive Property and the Properties of Fractions. 2 12 2 12 7 1 3 1 6 1 1 9x 3 8 14 9x 5 14 9x 9 x1 • Solve for x. 1 checks as a solution. The solution is 1. Occasionally, a value of the variable that appears to be a solution of an equation will make one of the denominators zero. In this case, that value is not a solution of the equation. Solve: Copyright © Houghton Mifflin Company. All rights reserved. HOW TO 4 2x 1 x2 x2 2x 4 1 x2 x2 2x 4 x 2 1 x2 x2 2x 4 x 2 x 2 1 x 2 x2 x2 x 2 1 1 x 2 x 2 2x 4 x 2 1 1 x2 1 x2 1 • The LCM is x 2. Multiply each side of the equation by the LCM. • Simplify using the Distributive Property and the Properties of Fractions. 1 2x x 2 4 2x x 2 x2 When x is replaced by 2, the denominators of Therefore, the equation has no solution. • Solve for x. 2x x2 and 4 x2 are zero. 316 Chapter 6 / Rational Expressions Example 1 Solve: You Try It 1 x 2 x4 x Solve: Solution x 3 x6 x Your solution The LCM is xx 4. x 2 x4 x x 2 xx 4 xx 4 x4 x 1 • Multiply by the LCM. 1 xx 4 x xx 4 2 1 x4 1 x 1 • Divide by common factors. 1 x2 x 42 x2 2x 8 • Simplify. Solve the quadratic equation by factoring. x2 2x 8 0 x 4x 2 0 x40 x20 x4 x 2 • Write in standard form. • Factor. • Principle of Zero Products Both 4 and 2 check as solutions. The solutions are 4 and 2. Example 2 Solve: You Try It 2 3x 12 5 x4 x4 Solve: Solution 5x 10 3 x2 x2 Your solution The LCM is x 4. 1 • Clear denominators. 1 x 4 3x x 4 12 x 45 1 x4 1 x4 1 3x x 45 12 3x 5x 20 12 3x 5x 8 2x 8 x4 1 • Solve for x. 4 does not check as a solution. The equation has no solution. Solutions on p. S17 Copyright © Houghton Mifflin Company. All rights reserved. 3x 12 5 x4 x4 3x 12 x 4 x 4 5 x4 x4 Section 6.5 / Solving Equations Containing Fractions 317 6.5 Exercises Objective A 1. To solve an equation containing fractions Can 2 be a solution of the equation 6x x1 x x2 4? Explain your answer. 2. After multiplying each side of an equation by a variable expression, why must we check the solution? For Exercises 3 to 35, solve. 3. 2x 5 1 3 2 2 4. x 1 1 3 4 12 5. x 1 x 1 3 4 4 6 6. 2y 1 y 1 9 6 9 6 7. 2x 5 1 x 3 8 4 8 4 8. 3x 4 1 5x 2 1 12 3 12 2 9. 6 2 2a 1 10. 12 3 3x 2 12. 6 3 4 3x 13. 2 16. 3 Copyright © Houghton Mifflin Company. All rights reserved. 15. 1 9 4 x 11. 9 2 2x 5 5 7 x 14. 3 12 7 x 17. 2 59 y 8 5 n 18. 6 3 11 x 19. 3 4 x2 x 20. 5 3 x3 x1 21. 2 3 3x 1 4x 1 22. 5 3 3x 4 1 2x 23. 3 2 2x 5 x1 318 24. Chapter 6 / Rational Expressions 4 2 5y 1 2y 1 27. 2 30. 3 a a3 a3 x 1 x 12 x5 33. x 6 3x x2 x2 25. 4x 5x 5 x4 x4 26. 2x 7x 5 x2 x2 28. x 4 3 x4 x4 29. x 8 x1 x2 31. 2x 3 x4 x1 32. 5 n 3n 8 n2 34. x 35. 8 2 1 y y2 6 2x x3 x3 APPLYING THE CONCEPTS 36. Explain the procedure for solving an equation containing fractions. Include in your discussion how the LCM of the denominators is used to eliminate fractions in the equation. 37. 3 1 2y 5 y 1 y 5 3 15 38. 3 1 a2 a 3 a 4 2 4 39. b2 1 3 b b 1 5 4 10 40. x 3 3 2 2x x 1 x 1 2x 1 41. x1 x2 3 2 x2 x 2 x 1 x2 42. y2 y1 1 2 y2 y 2 y 4 y1 2 Copyright © Houghton Mifflin Company. All rights reserved. For Exercises 37 to 42, solve.