Solving Rational Equations

1. Section 12.5 Solving Rational Equations

2. Examples Examples of rational equations in one variable Solving Rational Equations in One Variable Examples of Rational Equations of One Variable

3. Example Solution Solving Rational Equations in One Variable Solving a Rational Equation

4. Example Solution Continued 1 is not an excluded value Solving Rational Equations in One Variable Solving a Rational Equation

5. Solution Continued Check using “intersect” on a graphing calculator Example Solving Rational Equations in One Variable Solving a Rational Equation

6. Solution Check using “intersect” on a graphing calculator Solving Rational Equations in One Variable Solving a Rational Equation

7. 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 Example Solving Rational Equations in One Variable Solving a Rational Equation

8. Solution Excluded values are –2 and 5 Clear the denominator by multiplying by the LCD Solving Rational Equations in One Variable Solving a Rational Equation

9. Solution Solution Continued Excluded values are –2 and 5 Clear the denominator by multiplying by the LCD Solving Rational Equations in One Variable Solving a Rational Equation

10. Solution Solution Continued Excluded values are –2 and 5 Clear the denominator by multiplying by the LCD Solving Rational Equations in One Variable Solving a Rational Equation

11. Solution Solution Continued 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

12. Process 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

13. Example a Solution Solving Rational Equations in One Variable Finding Input Value of a Rational Function

14. Example Solution Continued a Solving Rational Equations in One Variable Finding Input Value of a Rational Function

15. Example Solution Continued Both results are in the domain of f Verify using a graphing calculator Solving Rational Equations in One Variable Finding Input Value of a Rational Function

16. Process 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

17. Clarification Solving Rational Equations verses Simplifying a Rational Expression Solving Rational Equations verses Simplifying Rational Expressions

18. 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. Using a Rational Model to Make Predictions About the Independent Variable Using a Rational Model to Make a Prediction

19. Solution Using a Rational Model to Make a Prediction Using a Rational Model to Make Predictions About the Independent Variable

20. Solution Solution Continued t = 2006. 3189 (year 5179) is not a reasonable answer Verify using a graphing calculator Using a Rational Model to Make Predictions About the Independent Variable Using a Rational Model to Make a Prediction