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

2. 6.3 – Simplifying Complex Fractions24
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
Outers over Inners
LCD: 63

6. 6.3 – Simplifying Complex Fractions
Outers over 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
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?
Time to sort one batch (hours)
Fraction of the job completed in one hour
Ryan
Mike
Together 2 3 x

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

16. 6.6 – Rational Equations and Problem Solving
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?
Time to Assemble one unit (hours)
Fraction of the job completed in one hour
Merry
Pippen
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
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?
Time to pump one basement (hours)
Fraction of the job completed in one hour
1st pump
2nd pump
Together 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.
Rate
Time
Distance
Motor-cycle
Car x t
450 mi t
x + 15
600 mi

23. 6.6 – Rational Equations and Problem Solving
Rate
Time
Distance
Motor-cycle
Car 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?
Rate
Time
Distance Up
Stream
Down
boat speed x t
x - 5
22 mi t
x + 5
42 mi

26. 6.6 – Rational Equations and Problem Solving
Rate
Time
Distance Up
Stream
Down
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 x y 3 5 9 13 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 x y 3 9 10 18 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