6.5 – Solving Equations w/ Rational Expressions

1. 6.5 – Solving Equations w/ Rational Expressions LCD: 20

2. 6.5 – Solving Equations w/ Rational Expressions LCD:

3. 6.5 – Solving Equations w/ Rational Expressions LCD: 6x

4. 6.5 – Solving Equations w/ Rational Expressions LCD: x+3

5. 6.5 – Solving Equations w/ Rational Expressions LCD:

6. 6.5 – Solving Equations w/ Rational Expressions Solve for a LCD: abx

7. 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

8. 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 sort a batch in 3 hours. If they work together, how fast can they sort one batch? 2 3 x

9. 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.

10. 6.6 – Rational Equations and Problem Solving Time to mow one acre (hours) Fraction of the job completed in one hour James Andy Together James and Andy mow lawns. It takes James 2 hours to mow an acre while it takes Andy 8 hours. How long will it take them to mow one acre if they work together? 2 8 x

11. 6.6 – Rational Equations and Problem Solving Time to mow one acre (hours) Fraction of the job completed in one hour James Andy Together 2 8 x LCD: 8x hrs.

12. 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

13. 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

14. 6.6 – Rational Equations and Problem Solving 60x LCD: hrs.

15. 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?

16. 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

17. 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)

18. 6.6 – Rational Equations and Problem Solving x(x + 15) x(x + 15) Motorcycle Car

19. 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

20. 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)

21. 6.6 – Rational Equations and Problem Solving (x – 5)(x + 5) (x – 5)(x + 5) Boat Speed

22. 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

23. 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

24. 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

25. 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.

26. 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

27. 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

28. 6.7 – Variation and Problem Solving Joint Variation If the ratio of a variable y to the product of two or more variables is constant, then y varies jointly as, or is jointly proportional, to the other variables. Therefore, if , then the number k is the constant of variation or the constant of proportionality. z varies jointly as x and y. x = 3 and y = 2 when z = 12. Find z when x = 4 and y = 5.

29. 6.7 – Variation and Problem Solving Joint Variation The volume of a can varies jointly as the height of the can and the square of its radius. A can with an 8 inch height and 4 inch radius has a volume of 402.12 cubic inches. What is the volume of a can that has a 2 inch radius and a 10 inch height? V varies jointly as h and . V = 402.12 cubic inches, h = 8 inches and r = 4 inches. Find V when h = 10 and r = 2.

30. Additional Problems

31. 6.5 – Solving Equations w/ Rational Expressions LCD: 15

32. 6.5 – Solving Equations w/ Rational Expressions LCD: x

33. 6.5 – Solving Equations w/ Rational Expressions LCD: Not a solution as equations is undefined at x = 1.

34. 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

35. 7.1 – Radicals Radical Expressions Finding a root of a number is the inverse operation of raising a number to a power. This symbol is the radical or the radical sign radical sign index radicand The expression under the radical sign is the radicand. The index defines the root to be taken.

36. 7.1 – Radicals Radical Expressions The above symbol represents the positive or principal root of a number. The symbol represents the negative root of a number.

37. 7.1 – Radicals Square Roots A square root of any positive number has two roots – one is positive and the other is negative. If a is a positive number, then is the positive square root of a and is the negative square root of a. Examples: non-real #

38. 7.1 – Radicals Rdicals Cube Roots A cube root of any positive number is positive. A cube root of any negative number is negative. Examples:

39. 7.1 – Radicals nth Roots An nth root of any number a is a number whose nth power is a. Examples:

40. 7.1 – Radicals nth Roots An nth root of any number a is a number whose nth power is a. Examples: Non-real number Non-real number

41. 6.4 – Synthetic Division

42. 6.4 – Synthetic Division Synthetic division will only work with linear factors with an one as the x-coefficient.