1. Lesson 8-1: Multiplying and Dividing Rational Expressions

2. Rational Expression
Definition: a ratio of two polynomial expressions

3. To Simplify A Rational Expression
1. Make sure both the numerator and denominator are factored completely!!!
2. Look for common factors and cancel
Remember factors are things that are being multiplied you can NEVER cancel things that are being added or subtracted!!!
3. Find out what conditions make the expression undefined and state them.

4. Examples: Simplify and state the values for x that result in the expression being undefined1. 2.

5. Examples Cont… Simplify3. 4. 5.

6. Operations with Rational Expressions
To Multiply Rational Expressions:
Factor and cancel where possible. Then multiply numerators and denominators
Define x-values for which the expression is undefined
To Divide Rational Expressions:
Rewrite the problem as a multiplication problem with the first expression times the reciprocal of the second expression. Then factor and cancel where possible. Multiply numerators and denominators

7. Examples: Simplify 6. 7. 8. 9.

8. Polynomials in Numerator and Denominator
Rules are the same as before…
1. Make sure everything is factored completely
2. Cancel common factors
3. Simplify and define x values for which the expression is undefined.

9. Examples: Simplify and define x values for which it is undefined
10.
11.

10. Examples:
12.
13.

11. Simplifying complex fractions
A complex fraction is a rational expression whose numerator and/or denominator contains a rational expression

12. To simplify complex fractions
Same rules as before
Rewrite as multiplication with the reciprocal
Factor and cancel what you can
Simplify everything
Multiply to finish

13. Examples:
14.
15.

14. Lesson 8-2: Adding and Subtracting Rational Expressions

15. Adding and Subtracting Rational Expressions
Finding Least Common Multiples and Least Common Denominators!
Use prime factorization
Example: Find the LCM of 6 and 4
6 = 2·3
4 = 22
LCM= 22·3 = 12

16. Find the LCM 1. 18r2s5, 24r3st2, and 15s3t 2. 15a2bc3, 16b5c2, 20a3c6
3. a2 – 6a + 9 and a2 + a -12
4. 2k3 – 5k2 – 12k and k3 – 8k2 +16k

17. Add and Subtract Rational Expressions
Same as fractions…
To add two fractions we find the LCD, the same things is going to happen with rational expressions

18. Examples: Simplify 5. 6. 7.

19. 9. 8.
10.

20. 11.
12.

21. 13.
14.

22. Lesson 8-3: Graphing Rational Functions

23. Definitions
Continuity: graph may not be able to be traced without picking up pencil
Asymptote: a like that the graph of the function approaches, but never touches (this line is graphed as a dotted line)
Point discontinuity: a hole in the graph

24. Vertical Asymptote How to find a Vertical Asymptote:
x = the value that makes the rational expression undefined
*Set the denominator of the rational expression equal to zero and solve.

25. Point Discontinuity How to find point discontinuity:
* Factor completely
* Set any factor that cancels equal to zero and solve. Those are the
x values that are points of discontinuity

26. Graphing Rational Functions
f(x) =

27. Graphing Rational Functions
f(x) =

28. Graphing Rational Functions
f(x) =

29. Graphing Rational Functions
f(x)=

30. Graphing Rational Functions
f(x) =

31. Graphing Rational Functions
f(x) =

32. Lesson 8-4: Direct, Joint, and Inverse Variation

33. Direct Variation
y varies directly as x if there is a nonzero constant, k, such that y = kx
*k is called the constant of variation
Plug in the two values you have and solve for the missing variable
Plug in that variable and the other given value to solve for the requested answer

34. Example If y varies directly as x and y = 12 when
x = -3, find y when x = 16.

35. Joint Variation
y varies jointly as x and z if there is a nonzero constant, k, such that y = kxz
* Follow the same directions as before

36. Example
Suppose y varies jointly as x and z. Find y when x = 8 and z = 3, if y = 16 when z = 2 and x = 5.

37. Inverse Variation
y varies inversely as x if there is a nonzero constant, k, such that xy = k or y= k x

38. Example
If y varies inversely as x and y = 18 when x = -3, find y when x = -11

39. Lesson 8-6: Solving Rational Equations and Inequalities

40. Let’s review some old skills
How do you find the LCM of two monomials
8x2y3 and 18x5
* Why do we find LCM’s with rational expressions?

41. Old Skills Cont…
What is it called when two fractions are equal to each other?
What process do we use to solve a problem like this?

42. To solve a rational equation:
1. Make sure the problem is written as a proportion
2. Cross Multiply
3. Solve for x
4. Check our answer

43. Examples
Solve

44. Let’s put those old skills to new use…
Solve

45. Solve . Check your solution.

47. Examples
Solve

48. Rational Inequalities
Lesson 8-6: Day #2
Rational Inequalities

49. Solving Rational Inequalities
Step 1: State any excluded values (where the denominator of any fraction could equal zero)
Step 2: Solve the related equation
Step 3: Divide a numberline into intervals using answers from steps 1 and 2 to create intervals
Step 4: Test a value in each interval to determine which values satisfy the inequality

50. Examples
Solve

51. Examples
Solve

52. Examples
Solve

53. Examples
Solve