1. Rational Functions and Variations Simplifying Rational Expression

2. Rational Function (7.1)

3. Inverse Variation(7.1)

4. Modeling(7.1)
A group of students in the chess club want to rent a bus to take them to a national chess competition. The bus cost $1500 to rent and can hold up to 60 people.
Write an equation that models the given information.
Cost per student if 30 students take the bus.
What is a reasonable domain and range?

5. Inverse Variation(7.1)

6. Direct Variation(7.1)

7. Car Repair(7.1)
The cost for labor at an auto repair shop is directly proportional to the time the mechanic spends working on the car. If the mechanic works on the car for five hours the labor cost is $325.
Write an equation that models the labor cost at the auto shop.
What is the labor cost for two hours of work?

8. Domain(7.1)

9. Example (7.1)

10. Simplify all fractions
#1 Rule (7.2)
Simplify all fractions

11. Methods (7.2)
Factor
Long Division
Synthetic Division

12. Factor (7.2)

13. Special case factoring
7(x – 5) (5 – x) Rewrite the denominator with the variable expression first 7(x – 5) 7(x – 5) 7(x – 5) (5 – x) (-x + 5) -1(x – 5) -7

14. Long Division

15. Performing Synthetic Division
Condenses the steps of long division without all the detail.
Performing Synthetic Division
The divisor (polynomial in the denominator) must be a binomial of the form x – c
Use only the coefficients of each term and not the variables

16. Example

17. Practice

18. Multiplying/Dividing(7.3)
The rules for fractions still apply when working with rational expressions! Multiplication 2ox • 6xy 18y2 10
Factor the numerator and the denominator
Divide out common factor
Rewrite as a single fraction in factored form

19. Dividing x + 3 • x – 7 = x + 3 x – 7 x – 4 x – 4
Keep
Change
Flip
Factor the numerator and denominator (if possible)
Divide out common factors and leave in factored form
x • x – = x + 3
x – x – x – 4

20. Adding/Subtracting

21. Practice

22. Practice
h h – 6 (h + 5) (h – 4) h – 4 a a – 3 a2 + 5a – 6 a2 + 8a + 12

23. Subtraction
Be careful! The numerator of the second fraction changes sign in subtraction. x2 + 5x - 3x + 8 (x + 3)(x + 4) (x + 3)(x + 4)

24. Complex Fraction