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.

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 5. 6. 7.

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 undefined8. 9.

10. Examples:
10.
11.

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:
12.
13.

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. a2 – 6a + 9 and a2 + a -12
3. 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 thing is going to happen with rational expressions

18. Examples: Simplify 4. 5. 6.

19. Examples 7. 8.

20. More Examples 9.
10.

21. And Two More
11.
12.

22. Lesson 8-3: Graphing Rational Functions

23. Definitions
Continuity: graph may not be able to be traced without picking up pencil
Asymptote: a line 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 8-5: Classes of Functions

40. General Equation Description of Graph
Contant
y = a
Horizontal line that crosses the y-axis at a
Direct Variation
y = ax
Line that passes through the origin and is neither horizontal nor vertical
Identity
Line that passes through a point (a,a) for any real number
Greatest Integer
A step function
Absolute Value
V-shaped graph
Quadratic
y = ax2 + bx + c
A parabola
Square Root
A curve that starts at a point and curves in only one direction (half a parabola)
Rational
A graph with one or more asymptotes and/or holes
Inverse Variation
A graph with 2 curved branches and 2 asymptotes

41. Identify the function represented by each graph

42. Identify the function represented by each graph

43. Identify the function represented by each equation
Identify the function represented by each equation. Then graph the equation. 5.

44. Identify and Graph

45. Identify and Graph
y = x – 4

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

47. 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?

48. 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?

49. To solve a rational equation:
1. Find the common denominator
2. Multiply both sides of the equation by the LCD
3. Solve for x
4. Check your answer

50. Examples
Solve

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

52. Examples
Solve Check your solution.

53. Examples

54. Examples
Solve

55. More Examples
Bill’s garden hose can fill the pool in 12 hours. His neighbor has a hose that can fill the pool in 15 hours. How long will it take to fill the pool using both hoses?

56. And Another Example
Joe can complete his yard work in 3 hours. If his son helps it will only take 2 hours working together. How long would the yard work take if his son was working alone?

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

58. Solving Rational Inequalities
1. State the excluded values.
2. Solve the related equation.
Divide a number line into intervals using answers from steps 1 and 2 to create intervals
Test a value in each interval to determine which values satisfy the inequality

59. Examples
Solve

60. Examples
Solve