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. Least Common Denominator LCD
Method 1
Least Common Denominator
LCD

4. Steps to solve Rational Equations
Find the LCD
Multiply every term on both sides of the equation by the LCD over 1(objective is to cancel out the denominators)
Solve for the variable
a. If it is a linear equation get variables on one side and constants on the other
b. If it is a quadratric set your equation = 0 and factor.

5. 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

6. Solving Rational Equations
Multiply both sides of the equation by the LCM of the denominators.
Least Common Multiple: Each factor raised to the greatest exponent.

7. Solve for x:
LCM =

8. Solve for x:
LCM =
(x + 1)(x)
(x + 1)(x)•
•(x + 1)(x)

9. Solve for x:
LCM = 2x
2x•
•2x

10. Solve for x: x•
• x

11. Solve when the denominators are the same, set the numerators to each other and solve
-2 is an extraneous solution.

12. Method 2
Cross Products

13. Cross products Short cut:
extraneous solution?

14. Cross products:

15. Cross products:

16. Cross products:

17. Method 3
Graphing

18. How to enter equation in graphing calculator
      
How to enter equation in graphing calculator
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.

19. Example 1: Solve

21. Algebraic Solution:

22. Example 2: Solve

23. 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.

24. Algebraic Solution:

25. 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).

26. 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.

27. 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.

28. 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

29. Car’s time =
Train’s time

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

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

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