1. Copyright © 2011 Pearson Education, Inc. Solving Rational Equations Section 12.5

2. Lehmann, Elementary and Intermediate Algebra, 1edSection 12.5Slide 2 Copyright © 2011 Pearson Education, Inc. Examples of rational equations in one variable Solving Rational Equations in One Variable Examples of Rational Equations of One Variable Examples

3. Lehmann, Elementary and Intermediate Algebra, 1edSection 12.5Slide 3 Copyright © 2011 Pearson Education, Inc. Solving Rational Equations in One Variable Solving a Rational Equation Example Solution

4. Lehmann, Elementary and Intermediate Algebra, 1edSection 12.5Slide 4 Copyright © 2011 Pearson Education, Inc. Solving Rational Equations in One Variable Solving a Rational Equation ExampleSolution Continued 1 is not an excluded value

5. Lehmann, Elementary and Intermediate Algebra, 1edSection 12.5Slide 5 Copyright © 2011 Pearson Education, Inc. Solution Continued Solving Rational Equations in One Variable Solving a Rational Equation Example Check using “intersect” on a graphing calculator

6. Lehmann, Elementary and Intermediate Algebra, 1edSection 12.5Slide 6 Copyright © 2011 Pearson Education, Inc. Solving Rational Equations in One Variable Solving a Rational Equation Solution

7. Lehmann, Elementary and Intermediate Algebra, 1edSection 12.5Slide 7 Copyright © 2011 Pearson Education, Inc. Solution Continued 2 is not a solution, it’s excluded because x - 2 = 0 would be in the denominator. Solution set is the empty set Solving Rational Equations in One Variable Solving a Rational Equation Example

8. Lehmann, Elementary and Intermediate Algebra, 1edSection 12.5Slide 8 Copyright © 2011 Pearson Education, Inc. Excluded values are –2 and 5 Clear the denominator by multiplying by the LCD Solving Rational Equations in One Variable Solving a Rational Equation Solution

9. Lehmann, Elementary and Intermediate Algebra, 1edSection 12.5Slide 9 Copyright © 2011 Pearson Education, Inc. Excluded values are –2 and 5 Clear the denominator by multiplying by the LCD Solving Rational Equations in One Variable Solving a Rational Equation SolutionSolution Continued

10. Lehmann, Elementary and Intermediate Algebra, 1edSection 12.5Slide 10 Copyright © 2011 Pearson Education, Inc. Since –2 is an excluded value; Solution is just 0 Plug 0 back into the original equation to verify the solution is correct. Solving Rational Equations in One Variable Solving a Rational Equation SolutionSolution Continued

11. Lehmann, Elementary and Intermediate Algebra, 1edSection 12.5Slide 11 Copyright © 2011 Pearson Education, Inc. To solve a rational equation in one variable, 1. Factor the denominator(s) if possible. 2. Identify any excluded values. 3.Find the LCD of all of the fractions. 4. Multiply both sides of the equation by the LCD, which gives a simpler equation to solve. 5. Solve the simpler equation. 6. Discard any proposed solutions that are excluded values. Solving Rational Equations in One Variable Solving a Rational Equations in One Variable Process

12. Lehmann, Elementary and Intermediate Algebra, 1edSection 12.5Slide 12 Copyright © 2011 Pearson Education, Inc. Solving Rational Equations in One Variable Finding Input Value of a Rational Function Example Solution

13. Lehmann, Elementary and Intermediate Algebra, 1edSection 12.5Slide 13 Copyright © 2011 Pearson Education, Inc. a Solving Rational Equations in One Variable Finding Input Value of a Rational Function ExampleSolution Continued

14. Lehmann, Elementary and Intermediate Algebra, 1edSection 12.5Slide 14 Copyright © 2011 Pearson Education, Inc. To solve a rational equation, clear the fractions in it by multiplying both sides of the equation by the LCD. To simplify a rational expression, do not multiply it by the LCD—the only multiplication permissible is multiplication by 1, usually in the form A/A, where A is a nonzero polynomial. Solving Rational Equations verses Simplifying Rational Expressions Solving Rational Equations verses Simplifying a Rational Expression Process

15. Lehmann, Elementary and Intermediate Algebra, 1edSection 12.5Slide 15 Copyright © 2011 Pearson Education, Inc. Solving Rational Equations verses Simplifying Rational Expressions Solving Rational Equations verses Simplifying a Rational Expression Clarification

16. Lehmann, Elementary and Intermediate Algebra, 1edSection 12.5Slide 16 Copyright © 2011 Pearson Education, Inc. Using a Rational Model to Make Predictions About the Independent Variable Using a Rational Model to Make a Prediction Example In Example 9 of Section 8.1, we found the model where P(t) is the percentage of Americans who are Internet users at t years since 1990 (see the table). Predict when 85% of Americans were or will be Internet users.

17. Lehmann, Elementary and Intermediate Algebra, 1edSection 12.5Slide 17 Copyright © 2011 Pearson Education, Inc. Using a Rational Model to Make Predictions About the Independent Variable Using a Rational Model to Make a Prediction Solution

18. Lehmann, Elementary and Intermediate Algebra, 1edSection 12.5Slide 18 Copyright © 2011 Pearson Education, Inc. Using a Rational Model to Make Predictions About the Independent Variable Using a Rational Model to Make a Prediction Solution t = 2006. 3189 (year 5179) is not a reasonable answer Verify using a graphing calculator Solution Continued