1. Angel, Elementary and Intermediate Algebra, 3ed 1 Rational Expressions and Equations Chapter 7

2. Angel, Elementary and Intermediate Algebra, 3ed 2 § 7.1 Simplifying Rational Expressions

3. Angel, Elementary and Intermediate Algebra, 3ed 3

4. 4 Rational Expressions is an expression of the form p/q where p and q are polynomials and q  0. A rational expression is an expression of the form p/q where p and q are polynomials and q  0. Examples: Whenever a rational expression contains a variable in the denominator, assume that the values that make the denominator 0 are excluded.

5. Angel, Elementary and Intermediate Algebra, 3ed 5 Signs of a Fraction Three signs are associated with any fraction: the sign of the numerator, the sign of the denominator, and the sign of the fraction. Changing any two of the three signs of a fraction does not change the value of the fraction - a + b + sign of the numerator sign of the denominator sign of the fraction - a b a -b = a b - =

6. Angel, Elementary and Intermediate Algebra, 3ed 6 Simplifying A rational expression is simplified or reduced to lowest terms when the numerator and denominator have no common factors other than 1. Examples:

7. Angel, Elementary and Intermediate Algebra, 3ed 7

8. 8 Simplifying Rational Expressions 1.Factor both the numerator and denominator as completely as possible. 2.Divide out any factors common to both the numerator and denominator.

9. Angel, Elementary and Intermediate Algebra, 3ed 9 Factoring a Negative 1 Remember that when –1 is factored from a polynomial, the sign of each term in the polynomial changes. Example: – 2x + 5 = – 1(2x – 5) = –(2x – 5)

10. Angel, Elementary and Intermediate Algebra, 3ed 10

11. Angel, Elementary and Intermediate Algebra, 3ed 11

12. Angel, Elementary and Intermediate Algebra, 3ed 12 § 7.2 Multiplication and Division of Rational Expressions

13. Angel, Elementary and Intermediate Algebra, 3ed 13 Multiplying Fractions Multiply. - 5 21 · 3 4 - 5 21 · 3 4 = - 5 21 · 3 4 1 7 - 5 7 · 1 4 == 28

14. Angel, Elementary and Intermediate Algebra, 3ed 14 Multiplying Rational Expressions 1.Factor all numerators and denominators completely. 2.Divide out common factors. 3.Multiply numerators together and multiply denominators together. Multiply

15. Angel, Elementary and Intermediate Algebra, 3ed 15

16. Angel, Elementary and Intermediate Algebra, 3ed 16 Dividing Two Fractions Divide. - 2 9  5 9 = 9  5 9 9 · 9 5 1 5 = 1 9 · 9 5 =

17. Angel, Elementary and Intermediate Algebra, 3ed 17 Dividing Rational Expressions Invert the divisor (the second fraction) and multiply Divide

18. Angel, Elementary and Intermediate Algebra, 3ed 18

19. Angel, Elementary and Intermediate Algebra, 3ed 19 Factor the x-box way Example: Factor 3x 2 -13x -10 -13x (3x 2 )(-10)= -30x 2 -15x 2x -10 -15x 2x 3x 2 x -5 3x +2 3x 2 -13x -10 = (x-5)(3x+2)

20. Angel, Elementary and Intermediate Algebra, 3ed 20 Factor the x-box way Example: Factor 2x 2 +3x -9 3x (2x 2 )(-9)= -18x 2 6x -3x -9 6x -3x 2x 2 x +3 2x -3 2x 2 +3x -9=(x+3)(2x-3)

21. Angel, Elementary and Intermediate Algebra, 3ed 21 Factor the x-box way Example: Factor 3x 2 -2x -5 -2x (3x 2 )(-5)= -15x 2 -5x 3x -5 -5x 3x 3x 2 3x -5 x +1 3x 2 -2x -5=(3x-5)(x+1)

22. Angel, Elementary and Intermediate Algebra, 3ed 22

23. Angel, Elementary and Intermediate Algebra, 3ed 23 § 7.3 Addition and Subtraction of Rational Expressions with a Common Denominator

24. Angel, Elementary and Intermediate Algebra, 3ed 24 Adding/Subtracting Fractions 7 12 = 5 2 + Add. 5 12 2 +

25. Angel, Elementary and Intermediate Algebra, 3ed 25 Common Denominators 1.Add or subtract the numerators. 2.Place the sum or difference of the numerators found in step 1 over the common denominator. 3.Simplify the fraction if possible. Subtract

26. Angel, Elementary and Intermediate Algebra, 3ed 26 Common Denominators a.)Add Example:

27. Angel, Elementary and Intermediate Algebra, 3ed 27 Common Denominators b.) Subtract Example:

28. Angel, Elementary and Intermediate Algebra, 3ed 28 Least Common Denominator 1.Factor each denominator completely. Any factors used more than once should be expressed as powers. 2.List all different factors that appear in any of the denominators. When the same factor appears in more than one denominator, write that factor with the highest power that appears. 3.The least common denominator (LCD) is the product of all the factors listed in step 2.

29. Angel, Elementary and Intermediate Algebra, 3ed 29 Least Common Denominator a.) Find the LCD: The LCD is (2x + 5)(x – 5). b.) The LCD is x(x + 1). c.) The LCD is 36w 5 z 4.

30. Angel, Elementary and Intermediate Algebra, 3ed 30 § 7.4 Addition and Subtraction of Rational Expressions

31. Angel, Elementary and Intermediate Algebra, 3ed 31 Unlike Denominators 1.Determine the LCD. 2.Rewrite each fraction as an equivalent fraction with the LCD. 3.Add or subtract the numerators while maintaining the LCD. 4.When possible, factor the remaining numerator and simplify the fraction.

32. Angel, Elementary and Intermediate Algebra, 3ed 32 Unlike Denominators a.) The LCD is w(w+2). Example:

33. Angel, Elementary and Intermediate Algebra, 3ed 33 Unlike Denominators b.) The LCD is 12x(x – 1). This cannot be factored any further. Example:

34. Angel, Elementary and Intermediate Algebra, 3ed 34 Unlike Denominators c.) The LCD is (x – 5)(x+2). Example:

35. Angel, Elementary and Intermediate Algebra, 3ed 35 § 7.5 Complex Fractions

36. Angel, Elementary and Intermediate Algebra, 3ed 36 Simplifying Complex Fractions A complex fraction is one that has a fraction in its numerator or its denominator or in both the numerator and denominator. Example:

37. Angel, Elementary and Intermediate Algebra, 3ed 37 Simplify by Combining Like Terms 1.Add or subtract the fraction in both the numerator and denominator of the complex fraction to obtain single fractions in both the numerator and the denominator. 2.Invert the denominator of the complex fraction and multiply the numerator by it. 3.Simplify further if possible.

38. Angel, Elementary and Intermediate Algebra, 3ed 38 a) b) Simplify by Combining Like Terms Simplify:

39. Angel, Elementary and Intermediate Algebra, 3ed 39 Simplify by Multiplying 1.Find the LCD of all the denominators appearing in the complex fraction. 2.Multiply both the numerator and the denominator of the complex fraction by the LCD. 3.Simplify further if possible.

40. Angel, Elementary and Intermediate Algebra, 3ed 40 Simplify by Multiplying a) LCD is 4b 3. 4b 3 b) LCD is y(x-y). y(x-y) Simplify:

41. Angel, Elementary and Intermediate Algebra, 3ed 41 § 7.6 Solving Rational Equations

42. Angel, Elementary and Intermediate Algebra, 3ed 42 Complex Fractions A rational equation is one that contains one or more rational (fractional) expressions. Example:

43. Angel, Elementary and Intermediate Algebra, 3ed 43 Solving Rational Equations 1.Find the LCD of all fractions in the equation. 2.Multiply both sides of the equation by the LCD. (Every term will be multiplied by the LCD.) 3.Remove any parentheses and combine like terms on each side of the equation. 4.Solve the equation. 5.Check the solution in the original equation.

44. Angel, Elementary and Intermediate Algebra, 3ed 44 Integer Denominators a) The LCD is 30. CHECK: Solve the equation:

45. Angel, Elementary and Intermediate Algebra, 3ed 45 Variable Denominators The LCD is 2z. Solve: CHECK: Whenever there is a variable in the denominator, it is necessary to check your answer in the original equation. If the answer obtained makes the denominator zero, that value is NOT a solution to the equation.

46. Angel, Elementary and Intermediate Algebra, 3ed 46 Variable Denominators The LCD is 2(x-3). Solve: CHECK: No solution

47. Angel, Elementary and Intermediate Algebra, 3ed 47 § 7.7 Rational Equations: Applications & Problem Solving

48. Angel, Elementary and Intermediate Algebra, 3ed 48 Work Problems Problems in which two or more people or machines work together to complete a certain task are referred to as work problems. part of task done by first machine part of task done by second machine 1 (one whole task completed) +=

49. Angel, Elementary and Intermediate Algebra, 3ed 49 Work Problems At the NCNB Savings Bank it takes a computer 4 hours to process and print payroll checks. When a second computer is used and the two computers work together, the checks can be printed in 3 hours. How long would it take the second computer by itself to process and print the payroll checks? Example: Continued.

50. Angel, Elementary and Intermediate Algebra, 3ed 50 Work Problems MachineRate of WorkTime Part of Task (rate x time) 1st 2nd A table helps to keep information organized. Example continued: Continued.

51. Angel, Elementary and Intermediate Algebra, 3ed 51 Work Problems MachineRate of WorkTime Part of Task (rate x time) 1st 2nd One job done Solve the equation. Example continued: Continued.

52. Angel, Elementary and Intermediate Algebra, 3ed 52 Work Problems The LCD is 4x. It would take the second computer 12 hours by itself. CHECK: Example continued: