ALGEBRA 2 X
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 |ASSIGNMENT |
|1 |8.2 MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS |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 |Operations Practice #1 |TBA |
|5 |Operations Practice #2 |TBA |
|6 |8.5 SOLVING RATIONAL EQUATIONS |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 |
|10 | |Enjoy the last day of winter. Spring is almost |
| |Unit 8 Test |here. |
Please be flexible as assignments may change.
Warm up: Simplify the following
1) [pic] 2) [pic] 3) [pic] 4) [pic]
When simplifying rational expressions…
Examples: Simplify and then state the values for x that make the expression undefined.
1) [pic] 2) [pic] 3) [pic]
[pic]__________ [pic]__________ [pic]__________
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.
1) [pic] 2) [pic] 3) [pic]
4) [pic]
Division is just like multiplication, except:_________________________________________________!
Examples: Divide. Assume that all expressions are defined.
1) [pic] 2) [pic] 3) [pic]
Mixed Practice:
1) [pic] 2) [pic] 3) [pic]
4) [pic] 5) [pic] 6) [pic]
7) [pic] * 8) [pic]
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 2) [pic] and 25x 3) [pic] and [pic]
4) [pic] and [pic] 5) [pic] and [pic]
Using example #3 from above, let’s add to rational expressions.
Also, state x-values that make the expressions undefined.
1) [pic]
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) [pic] 3) [pic]
Try on your own…
4) [pic] 5) [pic]
Subtracting is the same process, except you must be careful to ___________________ the negative!
1) [pic] 2) [pic]
Mixed Practice:
1. [pic] 2. [pic]
3. [pic] 4. [pic]
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. [pic]
[pic]
Complex Fractions: fractions that have a fraction in the numerator, denominator, or BOTH.
1. Simplify the following: [pic]
[pic] [pic]
2. [pic] 3. [pic]
4. [pic] 5. [pic]
6. [pic] 7. [pic]
8. [pic]
Let’s recap all that we have learned so far about rational expressions…
1) Multiplying:
2) Dividing:
3) Adding:
4) Subtracting:
5) Complex:
[pic]
[pic]
[pic]
[pic]
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)
[pic]
[pic]
[pic]
[pic]
[pic][pic]
[pic]
[pic]
There are two types of rational equations…
[pic] [pic]
1) [pic] 2) [pic]
* use either method * Multiply by LCD right away or
** Combine into 1 fraction then cross multiply
1. [pic] 2. [pic]
3. [pic]
4[pic] 5. [pic]
6. [pic]
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 |Work |Rate |
|Jason | |1 | |
|Lacy | |1 | |
|Together | |1 | |
Simplify and state the restrictions on x.
1. [pic] 2. [pic]
[pic] [pic]
Simplify. You do not need to state restrictions.
3. [pic] 4. [pic]
5. [pic] 6. [pic]
7. [pic] 8. [pic]
Solve the following equations. Don’t forget to check!
9. [pic] 10. [pic][pic]
11. [pic] 12. [pic]
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 [pic] hours. How long would it take Melissa to mow the lawn?
| |Time |Work |Rate |
|John | |1 | |
|Melissa | |1 | |
|Together | |1 | |
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 |Rate |Time |
|Upstream | |5 - x | |
|Downstream | |5 + x | |
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 |Work |Rate |
|Pipe A | |1 | |
|Pipe B | |1 | |
|Together | |1 | |
Directions: Use your formula sheet (if appropriate) to help you answer the following 11 questions.
[pic]
|Type of Variation |Equation Form |Ratio Form |Example |
| |[pic] |[pic] |[pic] |
|Direct | | | |
| |[pic] |[pic] |[pic] |
|Inverse | | | |
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 |2 |5 |9 |
|y |3 |6 |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.
40) Complete the table if y varies jointly as x and z.
|x |y |z |
|2 | |4 |
|5 |52.5 |7 |
| | | |
|1.5 |-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 |2 |10 |25 | |
|y |120 | |2400 |144 |
|z |5 |15 | |6 |
Answers!
1. [pic] 4. [pic] atmospheres.
2. [pic] 5. k = 12
|x |2 |10 |25 |2 |
|y |120 |1800 |2400 |144 |
|z |5 |15 |8 |6 |
3. [pic] miles; [pic] hours
-----------------------
Day 1: Multiplying and Dividing Rational Expressions
Day 8: Homework – PSSA Prep Worksheet #2 (After the Test)
Day 7: Unit Review
Day 9: Direct, Inverse, & Joint Variations
Day 6: Solving Rational Equations
Day 5: Classwork – Operations Practice #2
Day 4: Homework – PSSA Prep Worksheet #1 (Due Monday)
Day 4: Classwork – Operations Practice #1
Day 3: Complex Rational Expressions
Day 2: Adding and Subtracting Rational Expressions
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