Rational expressions and equations

1. Rational expressions and equations Chapter 6

2. 6.1 – rational expressions Chapter 6

3. Rational expressions What is a rational number? What might a rational expression be? A rational expression is an algebraic fraction with a numerator and a denominator that are polynomials. Examples:

4. Non-permissible values What value can x not have? For all rational expressions with variables in the denominator, we need to define the non-permissible values. These are the values for a variable that makes an expression undefined. In a rational expression, this is a value that results in a denominator of zero.

5. example For each rational expression, find its non-permissible values: a) When you have a denominator that is broken up into factors (numbers or expressions that multiply together), then you need to let each factor be equal to zero to find the non-permissible values: x = 0 (non-permissible value) 2x – 3 = 0  2x = 3  x = 3/2 (non-permissible value) Try it: b) (you will need to factor the denominator)

6. SIMPLIFING RATIONAL EXPRESSIONS Recall: To simplify rational expressions, we need to find any common factors in the numerator and denominator.

7. example Simplify, and state the non-permissible values: Need to factor the numerator and the denominator: Try it:

8. example Consider the expression . a) What expression represents the non-permissible values for x? b) Simplify the rational expression. c) Evaluate the expression for x = 2.6 and y = 1.2. Let 8x – 6y = 0  8x = 6y  x = 6y/8 = 3y/4  x ≠ 3y/4 Some examples, then of non-permission values are: (3/4, 1), (3/2, 2), (9/4, 3), and so on. b) • c) Make sure that the values are permissible: • 3y/4 = 3(1.2)/4) = 0.9 ≠ 2.6 • Value is fine. • You can use either expression (using the simplified version will be easier): • (4(2.6) + 3(1.2))/2 = 7 Recall: difference of squares

9. Pg. 317-321, #2, 4, 6, Independent Practice

10. 6.2 – Multiplying and dividing rational numbers Chapter 6

11. Multiplying rational numbers What is the rule for multiplying fractions? What are the non-permissible values?

12. example Multiply, and simplify the expression. Identify all non-permissible values. Cancel any common factors First, you need to factor as much as you can. What are the non-permissible values?

13. Try it

14. example Determine the quotient in simplest form. What’s the rule for dividing fractions? Why is –5 a non-permissible value?

15. Try it

16. example Simplify: Why did I factor out the –1 from (3 – 2m)?

17. Try it

18. Pg. 327-330, #2, 8, 12, 14, 15, 16, 19 Independent Practice

19. 6.3 – Adding and subtracting rational expressions Chapter 6

20. Adding and subtracting What’s the rule for adding and subtracting fractions? Adding and subtracting rational expressions is the same. If there is a denominators are the same, we need to add the numerators. If the denominators are not equal then we need to find a common denominator. For instance:

21. example What will our common denominator be? What will our common denominator be?

22. Try it

23. Pg. 336-340, #3, 5, 6, 7, 10, 14, 15, 25 Independent Practice

24. 6.4 – rational equations Chapter 6

25. riddle Diophantusof Alexandria is often called the father of new algebra. He is best known for his Arithmetica, a work on solving algebraic equations and on the theory of numbers. Diophantus extended numbers to include negatives and was one of the first to describe symbols for exponents. Although it is uncertain when he was born, we can learn his age when he died from the following facts recorded about him ... his boyhood lasted of his life; his beard grew after more; he married after more; his son was born 5 years later; the son lived to half his father’s age and the father died 4 years later. How many years did Diophantus live?

26. example Solve for z: First, factor all of the denominators. What will the common denominator be? Multiply each side by the common denominator. z ≠ 2, -2  So, z = 5 is an acceptable answer. What are the non-permissible values?

27. Try it Solve for y:

28. Example Solve the equation. What are some non-permissible values? The non-permissible values are 2 and -2.

29. example Solve the equation. What are some non-permissible values? Does k = –2 work? Why or why not?

30. Try it Solve the equation. What are some non-permissible values?

31. example Two friends share a paper route. Sheena can deliver the papers in 40 minutes, Jeff can cover the same route in 50 minutes. How long, to the nearest minute, does the paper route take if they work together? In one minute: Sheena can deliver 1/40th of the papers Jeff can deliver 1/50th of the papers So, while working together for x minutes we can say that: Check!

32. example In a dogsled race, the total distance was 140 miles. Conditions were excellent on the way to FlinFlon. However, bad weather caused the winner’s average speed to decrease by 6 mph on the return trip. The total time for the trip was 8.5 hours. What was the winning dog team’s average speed on the way to FlinFlon? Each half of the race is 70 miles. Lets call the speed that the winner was going on the way to FlinFlonx. Then, on the way back, he was travelling at a speed of x – 6. Recall that the formula for time is t = d/v On the way there the winner had a time of: On the way back, he had a time of: So, the total time (or 8.5 hours) is represented by the formula: Try solving for x.

33. Pg. 348-351, #1, 3, 5, 9, 13, 14, 19, 22 Independent Practice