2. Chapter 11 Sections 11.1, 11.3-11.5 Rational Expressions

3. § 11.1 Solving Proportions

4. Martin-Gay, Developmental Mathematics 3 Solve each proportion: ANSWER 1.) 2.) 16 24

5. Martin-Gay, Developmental Mathematics 4 Solve each proportion: ANSWER 1.) 2.) 28 4

6. Martin-Gay, Developmental Mathematics 5 Solve each proportion: ANSWER 1.) -10x - 10x -3 - x 2 + 48 2 ANSWER

7. Martin-Gay, Developmental Mathematics 6 Solve each proportion: - 5x ANSWERS

8. Martin-Gay, Developmental Mathematics 7 Solve the proportion for x. Solving Proportions Example

9. Martin-Gay, Developmental Mathematics 8 What is the value of “x” 4 x 8 3x – A – 6 B – 3 C 3 D 6 SOLUTION 4 x = 8 x– 3 Write original proportion. Cross products property 4(x – 3) = x 8 4x – 12 = 8x Simplify. Subtract 4x from each side. –12 = 4x Divide each side by 4. –3 = x

10. Martin-Gay, Developmental Mathematics 9 Example  Solve the equation below: - 3x 7 ANSWER

11. Martin-Gay, Developmental Mathematics 10 Solve each problem. +4x - 15 - 15 9 - 11x - 8

12. § 11.3 Simplifying Rational Expressions

13. Martin-Gay, Developmental Mathematics 12 Simplifying a Rational Expression 1) Completely factor the numerator and denominator. 2) Apply the Fundamental Principle of Rational Expressions to eliminate common factors in the numerator and denominator. Warning! YOU CAN ONLY ELIMINATE THINGS THAT ARE BEING MULTIPLIED!!! Simplifying Rational Expressions

14. Martin-Gay, Developmental Mathematics 13 Simplify the following expression. Simplifying Rational Expressions Example

15. Martin-Gay, Developmental Mathematics 14 Simplify the following expression. Simplifying Rational Expressions Example

16. Martin-Gay, Developmental Mathematics 15 Simplify the following expression. Simplifying Rational Expressions Example GCF Bottom

17. Martin-Gay, Developmental Mathematics 16 Simplify the following expression. Simplifying Rational Expressions Example GCF Top and Bottom

18. Martin-Gay, Developmental Mathematics 17 Simplify the following expression. Simplifying Rational Expressions Example

19. Martin-Gay, Developmental Mathematics 18 Simplify the following expression. Simplifying Rational Expressions Example

20. Martin-Gay, Developmental Mathematics 19 Simplify the following expression. Simplifying Rational Expressions Example GCF Bottom

21. Martin-Gay, Developmental Mathematics 20 Simplify the following expression. Simplifying Rational Expressions Example Factor Top and Bottom Use Sum and Product Method

22. Martin-Gay, Developmental Mathematics 21 Simplify the following expression. Simplifying Rational Expressions Example Factor Top by grouping BUT NOTHING WILL CANCEL OUT EXCEPT THE “Z”

23. Martin-Gay, Developmental Mathematics 22 Simplify the following expression. Simplifying Rational Expressions Example Factor Top by “difference of squares” Factor Bottom by “GCF”

24. Martin-Gay, Developmental Mathematics 23 Simplify the following expression. Simplifying Rational Expressions Example Factor Top by “difference of squares” Factor Bottom by “GCF”

25. Martin-Gay, Developmental Mathematics 24 Simplify the following expression. Simplifying Rational Expressions Example Factor Top by “Sum and Product”

26. Martin-Gay, Developmental Mathematics 25 Simplify the following expression. Simplifying Rational Expressions Example Factor Top by “GCF” Factor Bottom

27. Multiplying and Dividing Rational Expressions § 11.4

28. Martin-Gay, Developmental Mathematics 27 Multiplying Rational Expressions  Just remember: 1.) FACTOR IF POSSIBLE 2.) “TOP times TOP” and “BOTTOM times BOTTOM” 3.) THEN SIMPLIFY

29. Martin-Gay, Developmental Mathematics 28 Multiply the following rational expressions. Example Multiplying Rational Expressions

30. Martin-Gay, Developmental Mathematics 29 Multiply the following rational expressions. Multiplying Rational Expressions Example

31. Martin-Gay, Developmental Mathematics 30 Multiply the following rational expressions. Example Multiplying Rational Expressions

32. Martin-Gay, Developmental Mathematics 31 Multiply the following rational expressions. Example Multiplying Rational Expressions

33. Martin-Gay, Developmental Mathematics 32 JUST REMEMBER: Change it to multiplication of the reciprocal Dividing Rational Expressions

34. Martin-Gay, Developmental Mathematics 33 Divide the following rational expression. Dividing Rational Expressions Example

35. Martin-Gay, Developmental Mathematics 34 Multiply the following rational expressions. Example Multiplying Rational Expressions

36. Adding and Subtracting Rational Expressions with the Same Denominators § 11.5

37. Martin-Gay, Developmental Mathematics 36 Rational Expressions Remember how to add or subtract fractions?

38. Martin-Gay, Developmental Mathematics 37 Add the following rational expressions. Adding Rational Expressions Example

39. Martin-Gay, Developmental Mathematics 38 Subtract the following rational expressions. Subtracting Rational Expressions Example

40. Martin-Gay, Developmental Mathematics 39 Subtract the following rational expressions. Subtracting Rational Expressions Example

41. Martin-Gay, Developmental Mathematics 40 Subtract the following rational expressions. Subtracting Rational Expressions Example

42. Martin-Gay, Developmental Mathematics 41 Subtract the following rational expressions. Subtracting Rational Expressions Example

43. Martin-Gay, Developmental Mathematics 42 Subtract the following rational expressions. Subtracting Rational Expressions Example

44. Martin-Gay, Developmental Mathematics 43 Subtract the following rational expressions. Subtracting Rational Expressions Example

45. Martin-Gay, Developmental Mathematics 44 Subtract the following rational expressions. Subtracting Rational Expressions Example

46. Adding and Subtracting Rational Expressions with Different Denominators § 11.6

47. Martin-Gay, Developmental Mathematics 46 As stated in the previous section, to add or subtract rational expressions with different denominators, we have to change them to equivalent forms first. Unlike Denominators

48. Martin-Gay, Developmental Mathematics 47 Rewrite the rational expression as an equivalent rational expression with the given denominator. Equivalent Expressions Example

49. Martin-Gay, Developmental Mathematics 48 Adding or Subtracting Rational Expressions with Unlike Denominators 1)Find the LCD of all the rational expressions. 2)Rewrite each rational expression as an equivalent one with the LCD as the denominator. 3)Add or subtract numerators and write result over the LCD. 4)Simplify rational expression, if possible. Unlike Denominators

50. Martin-Gay, Developmental Mathematics 49 Add the following rational expressions. Adding with Unlike Denominators Example

51. Martin-Gay, Developmental Mathematics 50 Subtract the following rational expressions. Subtracting with Unlike Denominators Example

52. Martin-Gay, Developmental Mathematics 51 Subtract the following rational expressions. Subtracting with Unlike Denominators Example

53. Martin-Gay, Developmental Mathematics 52 Add the following rational expressions. Adding with Unlike Denominators Example