6.3 – Simplifying Complex Fractions

1. 6.3 – Simplifying Complex Fractions Complex Fractions Defn: A rational expression whose numerator, denominator, or both contain one or more rational expressions.

2. 6.3 – Simplifying Complex Fractions 24 24 LCD: 12, 8 LCD: 24 2 3

3. 6.3 – Simplifying Complex Fractions LCD: y y–y

4. 6.3 – Simplifying Complex Fractions LCD: 6xy 6xy 6xy

5. 6.3 – Simplifying Complex Fractions Outersover Inners LCD: 63

6. 6.3 – Simplifying Complex Fractions Outersover Inners

7. 6.5 – Solving Equations w/ Rational Expressions LCD: 20

8. 6.5 – Solving Equations w/ Rational Expressions LCD:

9. 6.5 – Solving Equations w/ Rational Expressions LCD: 6x

10. 6.5 – Solving Equations w/ Rational Expressions LCD: x+3

11. 6.5 – Solving Equations w/ Rational Expressions LCD:

12. 6.5 – Solving Equations w/ Rational Expressions Solve for a LCD: abx

13. 6.6 – Rational Equations and Problem Solving Problems about Numbers If one more than three times a number is divided by the number, the result is four thirds. Find the number. LCD = 3x

14. 6.6 – Rational Equations and Problem Solving Time to sort one batch (hours) Fraction of the job completed in one hour Ryan Mike Together Problems about Work Mike and Ryan work at a recycling plant. Ryan can sort a batch of recyclables in 2 hours and Mike can short a batch in 3 hours. If they work together, how fast can they sort one batch? 2 3 x

15. 6.6 – Rational Equations and Problem Solving Time to sort one batch (hours) Fraction of the job completed in one hour Ryan Mike Together Problems about Work 2 3 x LCD = 6x hrs.

16. 6.6 – Rational Equations and Problem Solving Time to Assemble one unit (hours) Fraction of the job completed in one hour Merry Pippen Together Pippen and Merry assemble Ork action figures. It takes Merry 2 hours to assemble one figure while it takes Pippen 8 hours. How long will it take them to assemble one figure if they work together? 2 8 x

17. 6.6 – Rational Equations and Problem Solving Time to Assemble one unit (hours) Fraction of the job completed in one hour Merry Pippen Together 2 8 x LCD: 8x hrs.

18. 6.6 – Rational Equations and Problem Solving Time to pump one basement (hours) Fraction of the job completed in one hour 1st pump 2nd pump Together A sump pump can pump water out of a basement in twelve hours. If a second pump is added, the job would only take six and two-thirds hours. How long would it take the second pump to do the job alone? 12 x

19. 6.6 – Rational Equations and Problem Solving Time to pump one basement (hours) Fraction of the job completed in one hour 1st pump 2nd pump Together 12 x

20. 6.6 – Rational Equations and Problem Solving LCD: 60x hrs.

21. 6.6 – Rational Equations and Problem Solving Distance, Rate and Time Problems If you drive at a constant speed of 65 miles per hour and you travel for 2 hours, how far did you drive?

22. 6.6 – Rational Equations and Problem Solving A car travels six hundred miles in the same time a motorcycle travels four hundred and fifty miles. If the car’s speed is fifteen miles per hour faster than the motorcycle’s, find the speed of both vehicles. x t 450 mi t x + 15 600 mi

23. 6.6 – Rational Equations and Problem Solving x t 450 mi t x + 15 600 mi LCD: x(x + 15) x(x + 15) x(x + 15)

24. 6.6 – Rational Equations and Problem Solving x(x + 15) x(x + 15) Motorcycle Car

25. 6.6 – Rational Equations and Problem Solving A boat can travel twenty-two miles upstream in the same amount of time it can travel forty-two miles downstream. The speed of the current is five miles per hour. What is the speed of the boat in still water? boat speed x t x - 5 22 mi t x + 5 42 mi

26. 6.6 – Rational Equations and Problem Solving boat speed x t x - 5 22 mi t x + 5 42 mi LCD: (x – 5)(x + 5) (x – 5)(x + 5) (x – 5)(x + 5)

27. 6.6 – Rational Equations and Problem Solving (x – 5)(x + 5) (x – 5)(x + 5) Boat Speed

28. 6.7 – Variation and Problem Solving Direct Variation: y varies directly as x(y is directly proportional to x), if there is a nonzero constant k such that The number k is called the constant of variation or the constant of proportionality

29. 6.7 – Variation and Problem Solving Direct Variation Suppose y varies directly as x. If y is 24 when x is 8, find the constant of variation (k) and the direct variation equation. direct variation equation constant of variation 13 5 3 9 9 15 27 39

30. 6.7 – Variation and Problem Solving Hooke’s law states that the distance a spring stretches is directly proportional to the weight attached to the spring. If a 56-pound weight stretches a spring 7 inches, find the distance that an 85-pound weight stretches the spring. Round to tenths. direct variation equation constant of variation

31. 6.7 – Variation and Problem Solving Inverse Variation: y varies inversely as x(y is inversely proportional to x), if there is a nonzero constant k such that The number k is called the constant of variation or the constant of proportionality.

32. 6.7 – Variation and Problem Solving Inverse Variation Suppose y varies inversely as x. If y is 6 when x is 3, find the constant of variation (k) and the inverse variation equation. direct variation equation constant of variation 10 18 9 3 6 2 1.8 1

33. 6.7 – Variation and Problem Solving The speed r at which one needs to drive in order to travel a constant distance is inversely proportional to the time t. A fixed distance can be driven in 4 hours at a rate of 30 mph. Find the rate needed to drive the same distance in 5 hours. direct variation equation constant of variation

34. Additional Problems

35. 6.5 – Solving Equations w/ Rational Expressions LCD: 15

36. 6.5 – Solving Equations w/ Rational Expressions LCD: x

37. 6.5 – Solving Equations w/ Rational Expressions LCD: Not a solution as equations is undefined at x = 1.

38. 6.6 – Rational Equations and Problem Solving Problems about Numbers The quotient of a number and 2 minus 1/3 is the quotient of a number and 6. Find the number. LCD = 6