# Rational Expressions and Equations

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Add/subtract and simplify:

1. 5 -4

___ + ________

c2d3 7cd2

36. 3x + 2 x - 2

_______ + ______

3x + 6 x2 - 4

50. 3(x - 2) 5(2x + 1) 3(x + 1)

_______ + ________ + ________

2x - 3 2x - 3 3 - 2x

18. 3 2

_________ _ _____________

2 2

12 + x - x x - 9

34. 4x - 6 7 - 2x

________ _ ________

x - 5 5 - x

40. a 5

_____________ - _________________________

2 2

a + 11a + 30 a + 9a + 20

Solve for x:

26. x + 1 x - 1

_______ _ _______ = 1

3 2

38. 4 2x 1

_____ + ________ = ____________

X - 3 2 - 9 x + 3

X

40. 5 30

______ - _________ = 1

y - 3 2

y - 9

51. Why is it especially important to check the possible solutions to a rational equation?

52. How can a graph be used to determine how many solutions an equation has?

8. Bobbi can pick a quart of raspberries in 20 min. Blanche can pick a quart in 25min.

How long would it take if Bobbi and Blanche worked together?

14. The speed of a freight train is 15 mph slower than the speed of a passenger train.

The freight train travels 390 mi in the same time that it takes the passenger train to

travel 480 mi. Find the speed of each train.

44. It takes 60 oz. of grass seed to seed 3000 ft of lawn. At this rate, how much would

be needed to seed 2 of lawn?

5000 ft

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#### Solution Preview

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Add. Simplify if possible.

5 -4

___ + ________

c2d3 7cd2

Put both over a common denominator:

5(7) + -4 (cd)

c2d3(7) 7cd2(cd)

35 + -4cd

7c2d3 7c2d3

Add the fractions:

35 - 4cd

7c2d3

36. 3x + 2 x - 2

_______ + ______

3x + 6 x2 - 4

Put both over a common denominator:

(3x + 2)(x2 - 4) + (x - 2)(3x + 6)

(3x + 6)(x2 - 4) (x2 - 4)(3x +6)

Add the fractions:

(3x + 2)(x2 - 4) + (x - 2)(3x + 6)

(x2 - 4)(3x +6)

Notice that x2 - 4 = (x - 2)(x + 2) and 3x + 6 = 3(x + 2).

Factor:

(3x + 2)(x - 2)(x + 2) + 3(x - 2)(x + 2)

3(x + 2)(x - 2)(x + 2)

(x - 2)(x + 2) [(3x + 2) + 3]

3(x + 2)(x - 2)(x + 2)

Cancel like terms:

[(3x + 2) + 3]

3(x + 2)

Simplify:

3x + 5

3(x + 2)

50. 3(x - 2) 5(2x + 1) 3(x + 1)

_______ + ________ + ________

2x - 3 2x - 3 3 - 2x

This is the same as above. Put over a common denominator, add, and simplify:

3(x - 2) 5(2x + 1) 3(x + 1)

_______ + ________ + ________

2x - 3 2x - 3 - (2x - 3)

3(x - 2) + 5(2x + 1) - 3(x + 1)

2x - 3

3x - 6 + 10x + 5 - 3x -3

2x - 3

10x - 4

2x - 3

2(5x - 2)

2x - 3

Subtract. Simplify if possible.

18. 3 2

_________ _ _____________

2 2

12 + x - x x - 9

Look at the denominator of the first fraction: -x2 + x + 12. That's the same as -(x2 -x -12), which can be factored as -(x - 4)(x + 3). The denominator of the ...

#### Solution Summary

This problem set has six problems involving simplifying rational expressions (that involve adding and subtracting fractions), three similar problems that involve solving rational equations, two discussion questions, and three word problems.