Name__________________________________________
Syllabus
Unit 8: Rational Expressions & Equations
We will most likely have a mini-quiz or two this unit.
LAST UNIT ‘TIL SPRING BREAK
DAY
TOPIC
1
8.2 MULTIPLYING AND DIVIDING
RATIONAL EXPRESSIONS
ASSIGNMENT
pg. 580: 1-29 ODDS (skip 17)
2
8.3 ADDING AND SUBTRACTING RATIONAL
EXPRESSIONS
pg. 588 # 7, 9, 10, 22, 26, 34, 35
3
8.3 COMPLEX RATIONAL EXPRESSIONS
pg. 588 # 1, 28-30, 43-45
4
5
6
Operations Practice #1
Operations Practice #2
8.5 SOLVING RATIONAL EQUATIONS
TBA
TBA
pg. 605 # 1-9 odd, 10, 11
7
8.1 DIRECT, INVERSE & JOINT VARIATIONS
(some hw problems done in class)
pg. 573 # 2-8 even, 9-11, 13-15
pg. 573 # 20-23, 31, 40, 41
8
UNIT REVIEW Day #1
Review Worksheet
9
UNIT Review Day #2
TBD
Unit 8 Test
Enjoy the last day of winter.
Spring is almost here.
10
Please be flexible as assignments may change.
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Day 1: Multiplying and Dividing Rational Expressions
Warm up: Simplify the following
1)
3 6
3
2)
x2  2 x
x
3)
x2  9
x3
4)
x 2  25
x2  6 x  5
3)
6 x2  7 x  3
3 x 2  x
When simplifying rational expressions…
Examples: Simplify and then state the values for x that make the expression undefined.
1)
4 x6
2x  6
x  __________
2)
x4
2
3 x  11x  4
x  __________
x  __________
Multiplying rational expressions is just like simplifying two at a time.
Any top can cancel with any ___________________ !!
Examples: Multiply. Assume that all expressions are defined.
x  2 4x  6

1)
2 x  3 x2  4
x 2  16
x2
 2
2) 2
x  4x  4 x  6x  8
3x x 2  9

3)
x  3 12 x3
x2  4x  3 x2  6x  8
 2
4)
x2  4
x  6x  8
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Division is just like multiplication, except:_________________________________________________!
Examples: Divide. Assume that all expressions are defined.
1)
x3
x3

2
x  2x 1 x 1
2)
x3
x

5 y 15 y 4
3)
x2  2 x  1
x2 1

x 2  3x  18 x 2  7 x  6
Mixed Practice:
1)
4x  8
x2  2x
2)
x2
1

x  4 3 x  12
3)
x 2  2 x  8 3x 2  10 x  8

9 x 2  16
x 2  16
4)
6 x3 y 2 2 xy 2

7z4
21z 2
5)
x 2  36
x 2  12 x  36
6)
4 x 2  3x 2 x  1

4x2 1
x
7)
1 x
*
x2 1
8)
4 1
4
Page 3 of 24
Day 2: Adding and Subtracting Rational Expressions
To add or subtract fractions, you must have a ____________________________________.
Let’s find a LCD (or LCM) for each of the following.
1) 12 and 18
4)
x
2
2) 5x 2 and 25x
 x  12  and  x  4 
5)
3)
x
2
 x  4
and  x  3
 25  and  x 2  10 x  25 
Using example #3 from above, let’s add to rational expressions.
Also, state x-values that make the expressions undefined.
1)
5  x  3
2x  x  4
5
2x




x  4 x  3  x  4  x  3  x  3 x  4 
To recap, here are the steps to adding rational expressions…
Step #1: Identify the __________
Step #2: Multiply each fraction by the _____________ ______________.
Step #3: Distribute (or FOIL) on each of the __________
Step #4: Add the tops and keep the bottoms the __________.
Step #5: State the values that make the expressions undefined (think ___________ = ____)
2)
7x
x2

x2  5x x  5
3)
2
x

x  4 x 3
Page 4 of 24
Try on your own…
4)
5)
Subtracting is the same process, except you must be careful to ___________________ the negative!
1)
2)
4x  3 2x  3

x2  9 x  3
Mixed Practice:
1.
2.
3.
4.
Page 5 of 24
Day 3: Complex Rational Expressions
Warm up: Simplify the following 4 problems. Be sure to state restrictions on the variables.
1.
2.
Multiply or divide. Write your answer in simplest form. Be sure to state restrictions on the variables.
3.
4.
x 2  25 x  5

x  3 x2  9
Complex Fractions: fractions that have a fraction in the numerator, denominator, or BOTH.
1
3
x
1. Simplify the following:
5
4
y
1 1

x y
2.
2 1

y x
3.
3
1
1
2y
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x 1
4. x  5
x6
x3
x
5. 2
3
x
12
6. x2 3
x 1
x2
1
7. x  5
x
2
x2
8. 3
3x  6
9
Let’s recap all that we have learned so far about rational expressions…
1) Multiplying:
2) Dividing:
3) Adding:
4) Subtracting:
5) Complex:
Page 7 of 24
Day 4: Classwork – Operations Practice #1
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Page 9 of 24
Day 4: Homework – PSSA Prep Worksheet #1 (Due Monday)
Directions: use the formula sheet below to help you with the practice PSSA test on the following 4 pages.
There are 22 multiple choice questions to help you prepare for the PSSA (coming up mid April)
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Day 5: Classwork – Operations Practice #2
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Day 6: Solving Rational Equations
There are two types of rational equations…
1)
1
7

x  5 2x
* use either method
1.
3.
x
8
 2
x
2)
1 2 17
 
8 x 8x
* Multiply by LCD right away or
** Combine into 1 fraction then cross multiply
2.
5
7
18

 2
x  3 x  4 x  x  12
k
k

2
k 5 k 2
Page 17 of 24
4
x
7

x 1 1  x
6.
5.
12
8
 2
x 1 x
6
3x

 3
x 2  5 x  66 x  11
Word Problems
7.
A kayaker spends an afternoon paddling on a river. She travels 3 miles upstream and 3 miles
downstream in a total of 4 hours. In still water, the kayaker can travel at an average speed of 2 miles per
hour. Based on this information, what is the average speed of the river’s current?
Distance
Rate
Time
Upstream
Downstream
8. Jason can clean a large tank at an aquarium in about 6 hours. When Jason and Lacy work together, they
can clean the tank in about 3.5 hours. About how long would it take Lacy to clean the tank if she worked
alone?
Time
Jason
Lacy
Together
Work
1
1
1
Rate
Page 18 of 24
Day 7: Unit Review
Simplify and state the restrictions on x.
1.
x3  3x 2  18 x
x 2  36
x  ____________
2.
15 x 2  240
25 x 2  200 x  400
x  ____________
Simplify. You do not need to state restrictions.
3.
x 2  x  12
2 x 2  9 x  18

2 x 2  11x  12
4 x  12
3
1

5. 2 x 1 x
3x
x
x
7.
5 1

x2 x
x6
4.
5 x  35 x 2  9 x  14

9 x2 1
30 x  10
6.
x3
x3
 2
x  2 x  4x  4
8.
5
x3
 2
x  3 x  8 x  15
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Solve the following equations. Don’t forget to check!
9.
5 2 1 5
  
2x 3 x 6
10.
11.
12
8
 2
x 1 x
12.
x
7

x 1 1  x
6
3x

 3
x  5 x  66 x  11
2
Write an equation and solve it to find the solution to each problem.
13. John can mow a lawn in 4 hours. When Melissa helps him, they can mow the lawn in 2 12 hours.
How long would it take Melissa to mow the lawn?
Time
John
Melissa
Together
Work
1
1
1
Rate
14. A boat travels 6 miles upstream in the same amount of time it can travel 10 miles downstream. In
still water the speed of the boat is 5 miles per hour. What is the speed of the current?
Distance
Upstream
Downstream
Rate
5-x
5+x
Time
Let x = the current of the water.
15. A water tank is filled by pipes from 2 wells. The first pipe can fill the tank in 4 days. The second
pipe can fill the tank in 6 days. How long will it take to fill the tank using both pipes?
Time
Pipe A
Pipe B
Together
Work
1
1
1
Rate
Page 20 of 24
Day 8: Homework – PSSA Prep Worksheet #2 (After the Test)
Directions: Use your formula sheet (if appropriate) to help you answer the following 11 questions.
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Day 9: Direct, Inverse, & Joint Variations
Type of
Variation
Equation Form
y  kx
k
Direct
y
Inverse
Ratio Form
y
x
k  xy
k
x
Example
y  45
x  5
y 3
x 8
Questions from HW:
2) If y varies directly as x, find an equation when y = 6 and x = 3.
9) If y varies inversely as x, find an equation when y = 2 and x = 7.
13) Determine whether each data set represents a direct variation, an inverse variation or neither.
x
y
2
3
5
6
9
4
A ______________________ variation is a relationship that contains both direct and inverse variation in one
problem. Directly will be in the ________________ and inversely will be in the _________________.
20) Medicine: The dosage d of a drug that a physician prescribes varies directly as the patient’s mass m,
and d = 100 mg when m = 55 kg. Find d to the nearest milligram when m = 70 kg.
22) Agriculture: The number of bags of soybean seeds N that a farmer needs varies jointly as the number
of acres a to be planted and the pounds of seed needed per acre p, and N = 980 when a = 700 acres and p =
70 lb/acre. Find N when a = 1000 acres and p = 75 lb/acre.
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40) Complete the table if y varies jointly as x and z.
x
2
5
y
1.5
52.5
z
4
7
-36
1.38
23
1. y varies directly with x, and x = 18 when y = 3. Find y when x = 66.
2. y varies jointly with x and z, and y = 200 when x = 4 and z = 20. Find x when y = 500 and z = 25.
3. Speed is inversely proportional to time. If I can reach my destination travelling at 50 mph for 2 hours,
how long would it take me at 65 mph?
4. The volume of a gas varies inversely with the pressure of the gas and directly with the temperature of the
gas. A certain gas has a volume of 10 L at a temperature of 300 K (“Kelvin,” an important unit of temp. in
Chemistry), and a pressure of 1.5 atmospheres (a unit of pressure). If the volume changes to 7.5 L and the
temperature increases to 350 K, what will the new pressure be?
5. Fill in the chart, given that y varies jointly with x and z.
x
y
z
2
120
5
10
25
2400
144
6
15
Answers!
1
1. y  kx, k  ; y  11
6
4. V 
2. y  kxz, k  2.5; x  8
5. k = 12
k
20
 1.54 hours
3. s  , k  100 miles; t 
t
13
x
y
z
kT
7
, k  .05; P   2.33 atmospheres.
P
3
2
120
5
10
1800
15
25
2400
8
2
144
6
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