1. Solving Rational Equations
To solve problems involving rational expressions using various methods

2. Rational Equation
An equation containing one or more rational expressions
When solving make sure that the answer does not make the denominator 0
Set the denominator equal to 0 and solve. This answer will cause the denominator to equal 0 and is called an extraneous answer.

3. Extraneous Solutions
When both sides of the equation are mult by a variable, the equation is transformed into a new equation and may have an extra solution.
Check each solution in the original rational equation
Make sure that your answer does not make the denominator 0

4. Cross Products

5. Cross products Short cut:
extraneous solution?

6. Cross products:

7. Cross products:

8. Cross products:

9. Graphing

10. How to enter equation Enter the left side of the equation into Y1.
      
How to enter equation
Enter the left side of the equation into Y1.
Enter the right side of the equation into Y2.
Use the INTERSECT option (2nd Trace #5).
The answer(s) is the x-value (the y-value is not used)
To get the answer as a fraction, go to the main screen, type x, the decimal answer will appear, hit math, then convert to fraction.

11. Example 1: Solve

13. Algebraic Solution:

14. Example 2: Solve

15. You can move the spider near x = 1 and zoom in to look for an intersection.  There is NO intersection at x = 1.
Division by zero is undefined.  x = 1 is NOT an answer. x = 1 is an extraneous root.

16. Algebraic Solution:

17. NOTE:
When working with rational equations, it may be difficult to "see" the intersection point if the viewing window is a small representation of the graph.  You may want to enlarge the viewing window by adjusting the WINDOW settings or by using ZOOM (#2 Zoom In).  Remember, when using Zoom In, hitting ENTER the first time only registers the function.  You must hit ENTER a second time to activate the Zoom In option.  You can quickly return to the 10 x10 viewing window by pressing ZOOM (#6 ZStandard).

18. Enrichment Example:
A car travels 500 miles in the same time that a train travels 300 miles. The speed of the car is 30 miles per hour faster than the speed of the train. Find the speed of the car and the train.

19. Remember the formula d=rt where:
r = rate of speed
d = distance
t = time
Since both vehicles travel the same amount of time, solve the formula for t.

20. Identify the variables that you are going to use.
Let r = speed of the train
How do you represent the speed of the car?
Let r+30 = speed of the car

21. Car’s time =
Train’s time

22. How would you solve this equation?
Cross-multiply

23. Make sure that you answer the question.
The car travels at a speed of 75mph
The train travels at a speed of 45 mph

24. Video Link(s) https://www.youtube.com/watch?v=9mskwxCDlao