1. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1 Section 6.6 Rational Equations Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1

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3. 3 Solving a Rational Equation A rational equation, also called a fractional equation, is an equation containing one or more rational expressions. The following is an example of a rational equation: Do you see that there is a variable in the denominator? This is a characteristic of many rational equations.

4. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 4 Solving a Rational Equation When adding or subtracting rational expressions, we find the LCD and convert fractions to equivalent fractions that have the common denominator. By contrast, when we solve rational equations, the LCD is used as a multiplier that clears an equation of fractions.

5. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 5 Solving a Rational EquationEXAMPLE Solve: SOLUTION Notice that the variable x appears in two of the denominators. We must avoid any values of the variable that make a denominator zero. This denominator would equal zero if x = 0. Therefore, we see that x cannot equal zero.

6. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 6 Solving a Rational Equation The denominators are 5x, 5, and x. The least common denominator is 5x. We begin by multiplying both sides of the equation by 5x. We will also write the restriction that x cannot equal zero to the right of the equation. CONTINUED This is the given equation. Multiply both sides by 5x, the LCD. Use the distributive property.

7. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 7 Solving a Rational EquationCONTINUED Divide out common factors in the multiplications. Multiply. Subtract. Add. Divide. The proposed solution, 8, is not part of the restriction. It should check in the original equation.

8. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 8 Solving a Rational EquationCONTINUED Check 8: This true statement verifies that the solution is 8 and the solution set is {8}. ? ? ? ? ? ? ? true

9. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 9 Solving a Rational Equation Solving Rational Equations 1) List restrictions on the variable. Avoid any values of the variable that make a denominator zero. 2) Clear the equation of fractions by multiplying both sides by the LCD of all rational expressions in the equation. 3) Solve the resulting equation. 4) Reject any proposed solution that is in the list of restrictions on the variable. Check other proposed solutions in the original equation.

10. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 10 Solving a Rational EquationEXAMPLE Solve: SOLUTION 1) List restrictions on the variable. This denominator would equal zero if x = 4. This denominator would equal zero if x = 3.5. The restrictions are

11. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 11 Solving a Rational Equation 2) Multiply both sides by the LCD. The denominators are x – 4 and 2x – 7. Thus, the LCD is (x – 4)(2x – 7). CONTINUED This is the given equation. Multiply both sides by the LCD. Simplify.

12. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 12 Solving a Rational Equation 4) Check the proposed solution in the original equation. Notice, there is no proposed solution. And of course, – 7 = – 20 is not a true statement. Therefore, there is no solution to the original rational equation. We say the solution set is, the empty set. CONTINUED This is the equation cleared of fractions. Use FOIL on each side. Subtract from both sides. Subtract 19x from both sides. 3) Solve the resulting equation.

13. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 13 Solving a Rational EquationEXAMPLE Solve: SOLUTION 1) List restrictions on the variable. By factoring denominators, it makes it easier to see values that make the denominators zero. This denominator is zero if x = -4 or x = 2. This denominator would equal zero if x = -4. The restrictions are This denominator would equal zero if x = 2.

14. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 14 Solving a Rational Equation 2) Multiply both sides by the LCD. The factors of the LCD are x + 4 and x – 2. Thus, the LCD is (x + 4)(x – 2). CONTINUED This is the given equation. Multiply both sides by the LCD. Use the distributive property.

15. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 15 Solving a Rational Equation 3) Solve the resulting equation. CONTINUED Simplify. This is the equation with cleared fractions. Use the distributive property. Combine like terms. Subtract x from both sides. Add 5 to both sides. Divide both sides by 3.

16. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 16 Solving a Rational Equation 4) Check the proposed solutions in the original equation. The proposed solution, 3, is not part of the restriction that Substitute 3 for x, in the given (original) equation. The resulting true statement verifies that 3 is a solution and that {3} is the solution set. CONTINUED

17. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 17 Objective #1: Example

18. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 18 Objective #1: Example

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27. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 27 Objective #1: ExampleCONTINUED

28. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 28 Objective #1: ExampleCONTINUED

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30. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 30 Solving a Rational EquationEXAMPLE Rational functions can be used to model learning. Many of these functions model the proportion of correct responses as a function of the number of trials of a particular task. One such model, called a learning curve, is where f (x) is the proportion of correct responses after x trials. If f (x) = 0, there are no correct responses. If f (x) = 1, all responses are correct. The graph of the rational function is shown on the next slide. Use the function to solve the following problem.

31. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 31 Solving a Rational EquationCONTINUED

32. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 32 Solving a Rational EquationCONTINUED SOLUTION How many learning trials are necessary for 0.5 of the responses to be correct? Identify your solution as a point on the graph. Substitute 0.5, the proportion of correct responses, for f (x) and solve the resulting rational equation for x. The LCD is 0.9x + 0.1. Multiply both sides by the LCD. Simplify.

33. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 33 Solving a Rational EquationCONTINUED Use the distributive property on the left side. Subtract 0.9x from both sides. Subtract 0.05 from both sides. Divide both sides by – 0.45. The number of learning trials necessary for 0.5 of the responses to be correct is 1. The solution is identified as a point on the graph at the beginning of the problem.

34. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 34 Objective #2: Example

35. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 35 Objective #2: ExampleCONTINUED

36. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 36 Objective #2: ExampleCONTINUED