Unit 4
Radical and Rational Equations
1.2:
1.3:
1.4:
2.1:
2.5:
2.6:
2.7:
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Unit 4
Exponents and Radicals
Factoring
Fractional Expressions
Solving Rational Equations
Solving Advanced Equations
Solving Inequalities
Solving Advanced Inequalities
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(1.2) Exponents and Radicals
A)
Simplify each expression.
1) (-3x2)4
2) (-2yz3)5
3) (503x100y90)0
B) Simplify each expression.
4) (3x)(5xy)
C)
3 4
5
5) (2x y )(-5xy z)
3x5 y
6)
9 xy 2
Re-write each expression without any negative exponents.
7) 5x-3
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8) 4m-1n2
Unit 4
9) ( 4mn2 ) - 3
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D) Solve for the possible values of each variable.
E)
10) x2 = 16
11) y2 = 49
12) w =
13) z =
16
Re-write in radical form.
1
14) x 2
F)
49
15) x

1
2
3
3
16) x 2
17) x 4
Re-write in exponential form.
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18)
3
19)
5
x2
20)
x9
22)
3
x7
23)
24x8
25)
5
24x7
26)
x
4
5x
G) Simplify as much as possible.
21)
24)
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32x5
5
15552 y 6 z12
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Open the TI-Nspire document Equivalent_or_Not.tns.
What does it mean for expressions to be equivalent? This
activity investigates expressions that are equivalent under
certain conditions on the variable.
Press / ¢ and / ¡ to navigate
Move to page 1.2.
through the lesson.
1. Find the value for each expression when
a. x= 2.
x2
= _____
x
x 2 = _____
|x| = _____
b. x= 4.
x2
= _____
x
x 2 = _____
|x| = _____
2. Based on your answers from question 1, predict the value for each expression when x = 15.
x2
= _____
x
x 2 = _____
|x| = _____
x 2 = _____
|x| = _____
3. Find the value for each expression when
a. x = 3.
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x2
= _____
x
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b. x = 8.
x2
= _____
x
x 2 = _____
|x| = _____
4. Based on your answers from question 3, predict the value for each expression when x = 20.
x2
= _____
x
x 2 = _____
|x| = _____
5. Find the value for each expression when x = 0.
a.
b.
c.
x2
x
x2
|x|
Two algebraic expressions that are equal for every substituted value of the variable chosen
from a set of numbers are said to be equivalent for that set of numbers.
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6. a. Is the expression
b. Is the expression
c. Is the expression
x2
equivalent to x for the set of positive real numbers? Why or why not?
x
x2
equivalent to x for the set of negative real numbers? Why or why not?
x
x2
equivalent to x for the set of real numbers? Why or why not?
x
7. Tom says that the expression
x 2 is equivalent to x for the set of real numbers. Do you agree? Why
or why not?
8. For what values of x are
Is 9. Is the expression
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x 2 and |x| equivalent? Explain your reasoning.
x equivalent to x for the set of real numbers? Why or why not?
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(1.2) Radicals Practice with Absolute Values
Simplify each radical expression. Do not assume variables are positive.
27)
80x5 y 2 z
28)
29) 6
a4b7c8d 9e12 f 13 g14
30)
32w14 h11
32)
31)
3
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x5 y 6 z 8
3
4
w7
w7
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33)
4
32c 6 d 10 e9 f 5
34)
5
160a4b6c7
1
 6a 2 b  5a 7 b   2
36) 
3 2 
2 
 15a b  2b  
1
2 2
35)
 4x 
37)
5k 4  2
3
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1
38)
Unit 4
 27 y 6  3
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39)
1
6 3
 27 y 
40)
8a 
9

2
3
1
12
2 4
41)  2 r 2s   3 r –1s0  


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Simplify. Do not assume positive variables.
42)
( x  4) 4
44)
4
46)
48)
3
43)
w8
45)
( x)7
47)
4
(m  3)12
3
54 xy 5
3
4
2y
x 7 y12 z 8
x8 y 6 z 3
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(1.2) Rationalizing Denominators
Rationalize the denominator.
1)
4)
7)
3
5
3
2)
8
6
3)
10
12
8
5)
4x
x
6)
4x
3
x
7
3
8)
125x5
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3
10 xy 4
7 x2
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(1.3) Polynomials
Find the degree of each polynomial.
2) 5x4 + 7x
1) 7
3) 12x3y2 – 7xy3
Simplify.
4) ( 4x2 + 5xy + 3y2 ) + ( 3x2 – 12xy + 4y2 )
5) ( 4x2 + 5xy + 3y2 ) - ( 3x2 – 12xy + 4y2 )
6) ( x + 5 )2
7) ( w -3k)( w + 3k)
8)
( 4x – w )3
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B. Factor:
9) 7x3 + 21x2
10) w2y3 – 25wy2
11) y2 – 9
12) m2 – 36
13) 8k2 – 200
Simplify:
14)
12 x3 w2  18 xw4
8 xw2
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Multiply:
15)
16)
17)
(x - y) (x2 +xy + y2 )
( x – 5 ) ( x2 + 5x + 25 )
( y + 3 ) ( y2 – 3y + 9 )
18) (3x + 2y ) ( 9x2 –6xy + 4y2 )
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F. Factor
19) 25w3 – 8
21)
20)
m2 + 5m + 6
22)
8k3 + 27m3
a2 + 3a – 10
23) 12x2 – 8x - 15
24) 50x2 + 45xy – 18y2
25) 5x3 +10x2 - 20x – 40
26) yx +2y – 3x – 6
27) x2 – 10x + 25 – 16y2
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Group Discussion:
What is the best way to factor each of the following expressions? Does it matter if you factor difference of
squares or difference of cubes first?
28) x6 - 26
29) y6 – 27z12
30) y6 – 5y3 + 4
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31) x12 – 1
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(1.4) Introduction to Fractional Expressions
Factor:
1) x2 – 4
2) x2 + 4x + 4
4) y2 – 9
5) z2 – 12z + 27
3) y3 – 27
6) 4z2 – 35z – 9
Simplify each fractional expression.
7)
x2  4
x2  4x  4
8)
9)
z 2  12 z  27
4 z 2  35 z  9
10)
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y 3  27
y2  9
3x 2  3xy  7 x  7 y
9 x 2  49
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Adding/Subtacting/Multiplying and Dividing Fractional Expressions
Perform the indicated operation, then simplify as much as possible.
11)
4
7

25 30
12)
4
7

2
x
xy
13)
( x  4)
x 2  9 x  18

x 2  7 x  6 x 2  x  12
14)
y2  4
y6
 2
2
y  4 y  12 y  36
15)
25 x 2  4 15 x 2  19 x  10

x2  2x  1
3x 2  8 x  5
16)
4
5

w3 w
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17)
4
5

w3 w 2
18)
4w
3w
5
 2
 2
2
w  9 w  5w  6 w  w  6
Fast-food workers:
19) A survey of high school students found that ½ of the female students had jobs and
2
/3 of the male students had jobs. It was also found that ½ of the female students, who
worked, worked in fast-food restaurants and 1/6 of the male students who worked,
worked in fast-food restaurants. If equal numbers of male and female students were
surveyed, then what fraction of the working students worked in fast-food restaurants?
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Simplify:
x y
2x  y
20)
x y
2x  y
1 1
( x  y )(  )
x y
21)
1 1
( x  y )(  )
x y
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22) At a Trigonometry University, ¼ of the undergraduate students commute, and 1/3 of the graduate students
commute. One-tenth of the undergraduate students drive more than 40 miles daily, and 1/6 of the graduate
students drive more than 40 miles daily. If there are twice as many undergraduate students as there are
graduate students, then what fraction of the commuters drive more than 40 miles daily.
23)
16 x 2  y 2
2 x y
{This next problem is a stretch, and won’t be on an assessment, but is interesting}
1
1
(1  x ) (2 x)  (2 x  3)   (1  x 2 ) 2 (2 x)
2
1
2 2
24)
1
2 2 2
[(1  x ) ]
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Rationalizing the numerator.
25)
Rationalize the numerator.
x h
x
Express as a polynomial. (Book says: Express as a sum in the form axr )
x2  4 x  6
26)
x4
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x2  4 x  6
27)
x
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(2.1) Practice Problems:
Solve.
3  5x 4  x

1)
3
8
3) 8 
5
7
 3
2x
2x
2)
13  2 x 5

4x  1 3
4) (2x - 5)2 =( x + 7)(4x + 3)
5) (4x+5)(6x-1) – 3x (8x+7) = 0
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(9.1) Group Extension:
Solving Systems Using any method.
1)
1
1
3
m k 
2
5
2
2
1
5
m k 
3
4
12
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2)
3
4

2
x 1 y  2
6
7

 3
x 1 y  2
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(2.2) Class Notes: Word Problems involving $$, Interest, testing, formulas:
1) Samuel has test scores of 80%, 71%, and 94%. What score on the 4th test will give Samuel
an exact average of 80%?
2) Caitlyn has an 88% average before a final exam. The exam is 1/5 of her final grade. What
minimum grade does she need on the final to secure a 90% for her final grade?
What if Caitlyn were satisfied with any B? (80% minimum)
What minimum grade would be needed on the final exam?
3) Casey has found her take home pay (this is also called the net pay) is $801.
What is her gross pay if she had 25% in deductions from her gross pay?
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4) An investment firm owned by Robert Winter and associates, has $100,000 to invest for Mason. It is
decided to invest it in two stocks, risky A and stable B. The expected annual rate of return, or simple
interest, for stock A is 9%, but there is some risk involved, and Mason does not wish to invest any more
than half his money in this stock. The annual rate of return on the more stable stock B is anticipated to
be 5%.
Determine whether there is a way, with the least amount of risk, of investing the money so that the
annual interest is $6,500 (that is the college tuition that he needs for his wonderful son, Max, to attend
his school of choice). He doesn’t want to lose any of the $100,000 because he has 3 more years of
college.
Determine whether there is a way of investing the money so that the annual interest is $8,000.
5) Nina is going to build a grain hopper (right cylinder with a right cone attached) such that
the radius of the base is 2 feet and the height of the cone is ½ of the height of the cylinder.
What value of the height of the cylinder, to the nearest tenth, will make the total volume of
the hopper 500 ft3?
6) If Aaron mixes a 100 ml of 4% cherry syrup solution with 400 ml of 9% cherry syrup solution, what
does he end up getting?
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Total amount of solution at the end:
Total amount of cherry syrup at the end:
% of cherry syrup at the end:
7) If Grace mixes 100 ml of 4% gold solution with 30 ml of pure gold, what does she
end up getting?
Total amount of solution at the end:
Total amount of gold at the end:
% of gold at the end:
8) If Lucy mixes 100 ml of 4% pseudoephedrine with 30 ml of water, what does she
end up getting?
Total amount of solution at the end:
Total amount of pseudoephedrine at the end:
% of pseudoephedrine at the end:
9) Master Mechanic, Allison Wexler, needs to determine the percent of antifreeze in a
radiator. A radiator contains 8 quarts of a mixture of water and antifreeze.
If 40% of the mixture is antifreeze, and we add 3 quarts of water, what is the percent, to the
nearest tenth, of antifreeze in the radiator?
10) A radiator contains 8 quarts of a mixture of water and antifreeze. If 40% of the mixture is
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antifreeze, how much of the mixture should Jaclyn drain and replace by pure antifreeze so that the
resultant mixture will contain 60% antifreeze?
11) Mr. Bouslog, a renowned chemist, has 10 ml of a solution that contains a 30%
concentration of acid. How many milliliters of pure acid must be added in order to
increase the concentration to 50%?
12) Two pumps are available for filling a gasoline storage tank. Pump A used alone,
can fill the tank in 5 hours, and pump B, used alone, can fill it up in 8 hours.
Joseph has decided to use both pumps simultaneously, but needs to know how
long will it take to fill the tank?
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13) Jacob leaves the Oasis, and travels to Iowa City for 3 hours going a constant rate 50 mph.
Cole leaves the same Oasis 30 minutes, after Jacob, going in the same direction, driving at
a constant rate, and arrives at the same time. At what rate will Cole need to travel to
arrive in Iowa City at the same time as Jacob arrives?
(You may assume they received no tickets and did not stop for bathroom breaks or food.)
14) Zoe leaves her house and walks 3 mph toward the high school. Her brother, Tyler, leaves their
house 20 minutes later, speed-walking 5 mph. How long, after Tyler leaves the house does it take him
to catch up to Zoe?
Given that their rates remain constant, and the high school is 2 miles away, will
Tyler beat Zoe to school?
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Group Practice
For the following problems, define any variables, write an equation or system of equations, and then
solve.
1. A Honda Accord costs $18,000 to purchase and $0.15 per mile to maintain. A Toyota Corolla costs
$16,500 to purchase and $0.21 per mile to maintain. How many miles must be driven for the total cost
of the 2 cars to be the same?
2. Dylan sells t-shirts and sweatshirts at a store in the mall. He has room for 510 shirts
all together. From experience Dylan knows that his profits will be greatest if he has 190
more t-shirts than sweatshirts. How many of each type of shirt should there be?
3. Sanford’s Gym offers 2 kick-boxing classes. There are currently 20 people regularly
going to the afternoon class, with ace instructor, Anna Zimmerman, and attendance is
increasing at 6 people per month because she is so amazing. There are currently 36 people
regularly attending the night class and attendance is increasing at a rate of 2 people per
month. When will the number in both classes be the same?
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For each of the following problems, define your variables; write the equations, and then graph to answer
the questions. Be sure to label your axes!
4. Nolan is comparing parking prices at a local concert. One option is a $7 entry fee plus $2
per hour. A second option is a $5 entry fee plus $3 per hour. When will the amount of money be the
same (break-even point) and how much money will that be? Which option do you think is better in
the long run? Explain your reasoning.
5. Oh no! Sydney has a hole in her pocket! She started with $9, but she is losing $2 a day. Amy is
benefiting from the situation because she only had $1 in her pocket and she is picking up Sydney’s $2
each day! On what day will they have the same amount of money and how much will that amount be?
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Systems of Inequalities
1. Jordan’s Restaurant ordered 200 flowers for Mother’s Day. They ordered carnations
at $1.50 each, roses at $5.75 each, and daisies at $2.60 each. They ordered mostly
carnations, and 20 fewer roses than daisies. The total order came to $589.50.
Set up a system of equations, then solve it. How many of each type of flower was
ordered?
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2) Graph each set of inequalities on the axes below.
y > 2x – 3
y<4–x
x > -3
3)
y<2
|x| < 3
y > -2
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LINE AR PROGRAMMING BACKGROUND
The first modules that we are developing for high schools will focus on linear
programming. The father of linear programming is George Dantzig, who developed
its foundation concepts between 1947 and 1949. During WWII, he worked on
developing various plans or proposed schedules of training, logistics supply and
deployment which the military calls programs. After the war he was challenged to
find an efficient way to develop these programs. He came to recognize that the
planning problem could be formulated as a system of linear inequalities. Dantzig was
the first to express the criterion for selecting a good or best plan as an explicit
mathematical function that we now call the objective function. All of this work would
have been of limited practical value without an efficient method, or algorithm, for
finding the optimal solution to a set of linear inequalities that maximizes (profit) or
minimizes (cost) for an objective function. He therefore proceeded to develop the
simplex algorithm which efficiently solves this problem. Interestingly, in 1939 the
Soviet mathematician and economist L.V. Kantorovich formulated and solved a linear
programming problem dealing with production planning. However, his work was
essentially unknown even in the Soviet Union and elsewhere for twenty years and
had no impact on the post WWII development of linear programming.
One group of academics who were excited by these developments were economists.
Several attendees at a first conference entitled Activity Analysis of Production and
Allocation went on to win Nobel prizes in economics with their work drawing on linear
programming to model fundamental economic principles.
The first problem Dantzig solved, much to the chagrin of his wife, was a minimum
cost diet problem that involved the solution of nine equations (nutrition
requirements) with seventy-seven decision variables. The National Bureau of
Standards supervised the solution which took 120 man days using hand operated
desk calculators. (His wife rejected the minimum cost diet as boring). Nowadays, a
standard personal computer could handle this problem under a second. EXCEL
spreadsheet software includes as a standard addition a module called “solver,” which
includes a linear programming solver.
As mainframe computers became available in the 50’s and grew more and more
powerful, the first major users of the simplex algorithm to solve practical problems
were the petroleum and chemical industries. One use was to minimize the cost of
blending gasoline to meet certain performance and content criterion. The field of
linear programming grew exponentially and led to the development of non-linear
programming in which inequalities and/or objective functions are non-linear
functions. Another extension is called integer programming (combinatorics) in which
the variables must take on only integer values. These disciplines are collectively
called mathematical programming.
©Industrial and Manufacturing Department –Wayne State University – August 18, 1997
Webmaster: Azriel Chelst HTML Master Sculptors Inc. (azriel@mie.eng.wayne.edu)
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A CASE WRITE UP:NABISCO BRANDS
When you eat the inside of an Oreo cookie or munch on a Ritz cracker, you
probably don’t realize that the production of the cookie you ate was planned with
mathematical programming. Production in the biscuit Division of Nabisco involved
two key operations, baking and secondary operations. In baking, raw materials are
fed into an oven. Secondary operations include sorting, packaging and labeling
finished products.
Scheduling and operation of bakeries is a difficult task. Each oven is able to
produce many but not all products. The efficiency of the ovens varies. The
secondary facilities at one site can be shared by several ovens in operation at the
same time. Production must be planned to keep the manufacturing and
transportation costs as low as possible. The key questions that are routinely
addressed with a mathematical model are:
Where should each product be produced?
How much of each product should be assigned to each oven?
From where should each product be shipped to each customer?
As new products are developed where should new plants be built?
What facilities should be placed in these plants?
One interesting problem involved the study of the differences between “snug” pack
vs. “dump” packs. In a traditional “dump” pack, the crackers are loose inside the
box. With the “snug” pack, crackers are stacked in three or more columns and
each column is wrapped separately. The model was used to plan the equipment
changeover for the different locations to convert to “snug” packaging.
A realistic problem at Nabisco could involve 150 products, 218 facilities, 10 plants,
and 127 customer zones. A problem this size involved over 44,000 decision
variables and almost 20,000 constraints. These problems were routinely solved in
1983 on an IBM 3033 computer in under 60 CPU seconds.
Brown, G.G., G. W. Graves and M.D. Honczarenko, “Design and Operation of a Multicommodity Production/Distribution
system Using Primal Goal Decomposition,” Management Science, Vol. 33, No. 11 (1987) pp. 1469 – 1480.
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LINEAR PROGRAMMING
Solving the Parking Problem
Example: The available parking area of a parking lot is 600 m2. A car requires 6 m2, and a bus requires
30 m2 of space. The attendant can handle no more than 60 vehicles. The parking fees are $2.50 for cars
and $7.50 for buses. How many of each type of vehicle should Daniel Barry, the attendant, accept to
maximize income? What is the maximum income?
120
100
80
60
40
20
20
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80
100
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Algebra II Trigonometry
Test Strategy
1. Matthew arrives late to Algebra class one day when there is a test! He only has 45 minutes to
complete the test. The test has 2 short answer questions and 30 multiple choice questions. Each correct
short answer question is worth 20 points and each multiple choice question is worth 2 points. Matthew
knows that it takes him 15 minutes to answer a short answer question, and 1 minute to answer a multiple
choice question. Assume that Matthew studied really hard for the test, and that each question he
answers he gets correct!
HOW MANY OF EACH TYPE OF QUESTION SHOULD HE ANSWER TO GET THE MAXIMUM
POSSIBLE POINTS ON THE TEST?
A. Write the inequality that describes “the number of short answer questions must be between 0 and
2 (including 0 and 2)
B. Write the inequality that describes “the number of multiple choice questions must be
between 0 and 30 (including 0 and 30)
C. Write the inequality that describes “The total time spent on the test must be less than or
equal to 45 minutes.”
Think about how much time Matthew spends on each type of problem.
D. What is the function that we are trying to maximize?
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Graph the inequalities from A, B, C, and D on the coordinate plane below:
Where are the corner points/vertices?
Which corner point gets the maximum value in the function from part E?
What is the answer to the question from the beginning?
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Practice: Linear Programming
1) Brooke owns a nutrition center that sells health food to mountain climbing teams.
The Trailblazer mix package contains one pound of corn cereal mixed with four
pounds of wheat cereal and sells for $9.75. The Frontier mix package contains two
pounds of corn cereal mixed with three pounds of wheat cereal and sells for $9.50.
The center has 60 pounds of corn cereal and 120 pounds of wheat cereal available.
How many packages of each mix should the Brooke sell to maximize her income?
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2) Jenna, an office manager at Deerfield High School, is purchasing new file cabinets.
The office has 60 square feet of floor space available, and $600 to spend.
Cabinet A requires 3 square feet of floor space, holds 12 cubic feet, and costs $75.
Cabinet B requires 6 square feet of floor space, holds 18 cubic feet, and costs $50.
How many of each kind of cabinet should Jenna buy to maximize the storage capacity?
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3) A manufacturer of a lightweight mountain tent makes a standard model, called the
‘Esgar’, and an expedition model, nicknamed the ‘Meyerhoff.’ Each ‘Esgar’ requires
one labor hour from the cutting department and 3 labor hours from the assembly
department. Each Meyerhoff tent requires 2 labor hours from the cutting department
and 4 labor hours from the assembly department. The maximum labor hours available
per week in the cutting and assembly departments are 32 and 84, respectively. In
addition, the distributor, because of demand, will not take more than 12 Meyerhoff tents
per week. If the company makes profit of $50 on each Esgar tent and $80 on each
Meyerhoff tent, how many tents of each type should be manufactured each week to
maximize the weekly profit?
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4) Skyler Mark, a famous biologist, is developing two new strains of bacteria. Each sample of
Type I bacteria produces four new viable bacteria, and each sample of Type II produces
three new viable bacteria. Altogether, at least 240 new viable bacteria must be produced. At
least 30, but not more than 60, of the original samples must be Type I. Not more than 70 of
the samples can be Type II. A sample of Type I costs $5.00 and a sample of Type II costs
$7.00. How many samples of Type II bacteria should be used to minimize cost?
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5) Meshi is going to make and sell bread. A loaf of Irish soda bread is made with 2 cups of
1
flour and cup of sugar. Kugelhopf cake is made with 4 cups of flour and 1 cup of sugar.
4
Meshi will make a profit of $4 on each loaf of Irish soda bread and a profit of $3 on each
Kugelhopf cake. She has 16 cups of flour and 3 cups of sugar.
a. How many of each kind of bread should Meshi make to maximize the profit?
b. What is the maximum profit?
6
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Linear Programming Practice
1) Mrs. Cotton wants to crochet beach hats and baby afghans for a church fund-raising bazaar.
She needs 7 hours to make a hat and 2 hours to make an afghan and she has 33 hours
available. She wants to make no more than 9 items and no more than 7 afghans. The bazaar
will sell the hats for $12 each and the afghans for $5 each. How many of each should she
make to maximize the income for the bazaar?
2) Javier Montano, an office manager needs to buy new filing cabinets. Cabinet A costs $7, takes up 7
square feet of floor space, and holds 7 cubic feet of files. Cabinet B costs $12, takes up 9 square feet,
and holds 13 cubic feet. He has only $128 to spend and the office has room for no more than 110 square
feet of cabinets. He wants to have at least one of each type of cabinet.
How many of each can he buy to maximize storage capacity?
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3) Ryan is bringing items to sell at a flea market, where he plans to sell televisions at $125 each
and DVD players at $100 each. Due to space limitations he can only store at most 150 items
for the day. However, because more people already own televisions, Ryan knows that the
number of DVD sales must at least match the number of television sales. How many of each
item should Ryan bring to the flea market to maximize his sales?
4) A candy company, Stecher Sweets, has 125 pounds of cashews and 150 pounds of peanuts
which can be for $7 per pound. The economy mix is one-third cashews and two-thirds
peanuts and sells for $5.90 per pound. How many pounds of each mix should be prepared
for maximum revenue?
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5) Dr. Montano has told a sick patient to take vitamin pills. The patient needs at least 30 units of vitamin
A, at least 8 units of vitamin B, and at least 32 units of vitamin C. The red vitamin pills cost $0.10 each
and contain 6 units of A, 1 unit of B, and 3 units of C. The blue vitamin pills cost $0.20 each and contain
3 units of A, 1 unit of B, and 7 units of C. How many pills should the patient take each day to minimize
costs?
6) The Sanford Class Ring Company designs and sells two types of rings: the VIP and the WOW. They
can produce up to 24 rings each day using up to 60 total hours of labor. It takes 3 hours to make one VIP
ring and 2 hours to make one WOW ring. How many of each type of ring should be made daily to
maximize the company's profit, if the profit on a VIP ring is $40 and on an WOW ring is $35?
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7) Suppose that Seth Cole desires 46 grams of protein and 38 grams of dietary fiber daily.
One serving of kidney beans has 8 grams of protein and 6 grams of dietary fiber. One
serving of refried pinto beans has 6 grams of protein and 6 grams of dietary fiber. If a
serving of kidney beans costs $0.45 and a serving of refried pinto beans costs $0.35, then
how many servings of each should Seth eat to minimize cost and still meet his
requirements?
Answers:
1. 3 hats, 6 afghans
3. 75 TV’s, 75 DVD players
6. 12 VIP, 12 WOW
2. 8 cabinet A, 6 cabinet B
4. 200 lbs deluxe, 75 lbs economy
7.
5. 6 red, 2 blue
7
servings pinto beans, 4 servings kidney beans
3
(2.3): Quadratic Equations
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Try to check using a different method.
1) Solve: x2 + 10x + 3 = 0
2) Solve: 4x2 – 24x = 10
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3) p. 84 # 63: The boundary of a city is a circle of diameter 5 miles. As shown in the Figure
(A,B and P are on the circle), a straight highway runs through the center of the city from
A to B. The highway department, headed by ace-architect, Michael Shklyar, is planning
to build a 6-mile long freeway from A to point P and then to B. Find the approximate
distance from A to P (nearest tenth).
A
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B
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4) p. 84 # 55: Mr. Utterback throws a baseball straight upward with an initial speed of
64 ft/sec. (He is lying on the ground to throw it) The number of feet, s, above
the ground after t seconds is given by the equation s = -16t2 + 64t.
a)
When will the baseball be 48 feet above the ground?
b)
When will it hit the ground?
c) What is the highest the ball will go and when will this happen?
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Summarize:
True or False? Justify your reasoning.
1) If a quadratic equation can be factored and each factor contains only real numbers,
then there cannot be an imaginary solution.
2) If a quadratic equation cannot be factored, then it will have at least 1 imaginary
solution.
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Solve and make a sketch of the solutions.
3)
a) -4n2 + 6n – 16 = -5n2
b) 8a2 + 6a = -5
c) x2 - 12x + 23 = 0
d) x2 = -4x - 4
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Finding Your Roots
March 7, 2010, 5:30 pm
http://opinionator.blogs.nytimes.com/2010/03/07/finding-your-roots/
By STEVEN STROGATZ
For more than 2,500 years, mathematicians have been obsessed with solving for x. The story of their
struggle to find the “roots” — the solutions — of increasingly complicated equations is one of the great
epics in the history of human thought.And yet, through it all, there’s been an irritant, a nagging little
thing that won’t go away: the solutions often involve square roots of negative numbers. Such solutions
were long derided as “sophistic” or “fictitious” because they seemed nonsensical on their face.
Until the 1700s or so, mathematicians believed that square roots of negative numbers simply couldn’t
exist. They couldn’t be positive numbers, after all, since a positive times a positive is always positive,
and we’re looking for numbers whose square is negative. Nor could negative numbers work, since a
negative times a negative is, again, positive. There seemed to be no hope of finding numbers which,
when multiplied by themselves, would give negative answers.
We’ve seen crises like this before. They occur whenever an existing operation is pushed too far, into a
domain where it no longer seems sensible. Just as subtracting bigger numbers from smaller ones gave
rise to negative numbers and division spawned fractions and decimals, the free-wheeling use of square
roots eventually forced the universe of numbers to expand…again.
Historically, this step was the most painful of all. The square root of –1 still goes by the demeaning
name of i, this scarlet letter serving as a constant reminder of its “imaginary” status.
This new kind of number (or if you’d rather be agnostic, call it a symbol, not a number) is defined by the
property that i2 = –1.
It’s true that i can’t be found anywhere on the number line. In that respect it’s much stranger than zero,
negative numbers, fractions or even irrational numbers, all of which — weird as they are — still have
their place in line.
But with enough imagination, our minds can make room for i as well. It lives off the number line, at
right angles to it, on its own imaginary axis. And when you fuse that imaginary axis to the ordinary
“real” number line, you create a 2-D space — a plane — where a new species of numbers lives.
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These are the “complex numbers.” Here complex doesn’t mean complicated; it means that two types of
numbers, real and imaginary, have bonded together to form a complex, a hybrid number like 2 + 3i.
Complex numbers are magnificent, the pinnacle of number systems. They
enjoy all the same properties as real numbers — you can add and subtract them, multiply and divide
them — but they are better than real numbers because they always have roots. You can take the square
root or cube root or any root of a complex number and the result will still be a complex number.
Better yet, a grand statement called The Fundamental Theorem of Algebra says that the roots of any
polynomial are always complex numbers. In that sense they’re the end of the quest, the holy grail. They
are the culmination of the journey that began with 1.
You can appreciate the utility of complex numbers (or find it more plausible) if you know how to
visualize them. The key is to understand what multiplying by i looks like.
Suppose we multiply an arbitrary positive number, say 3, by i. The result is the imaginary number 3i.
So multiplying by i produces a rotation counterclockwise by a quarter turn. It takes an arrow of length 3
pointing east, and changes it into a new arrow of the same length but now pointing north. Electrical
engineers love complex numbers for exactly this reason. Having such a compact way to represent 90degree rotations is very useful to them when working with alternating currents and voltages, or with
electric and magnetic fields, because these often involve oscillations or waves that are a quarter cycle
(i.e., 90 degrees) out of phase.
In fact, complex numbers are indispensable to all engineers. In aerospace engineering they eased the
first calculations of the lift on an airplane wing. Civil and mechanical engineers use them routinely to
analyze the vibrations of footbridges, skyscrapers and cars driving on bumpy roads.
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(2.4): Complex Numbers
Write a simplified expression for each term.
1) i100
2) i101
3) i83
4) i71
5) i64
6) i30
7) Find the value of x and y if: 8 + 3i = x – yi
8) Solve: x2 = - 16
9) Simplify: (4 + 7i) – (5 – 2i)
10) Simplify: (4 + 7i)(5 – 2i)
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11)
4  7i
5  2i
12)
3  121
7  36
{The text leaves answer in a + bi form}
Solve the equations for the unknown variable.
13)
x4 + 6x2 + 8 = 0
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14) x4 = 81
Page 93
15) x4 – 5x2 + 7 = 0
16) x3 = 64
17) x6 = 64
Summarize: Does the real number property of:
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a
b  ab apply to imaginary numbers?
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(2.5): Other Types of Equations
Practice: Solve.
1)
2)
4
2 x2 1  x
3 x  x  3
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3) 2 y  y  3  5  y
2
3
3
4
5) w  8
4) x  16
6)
9m  4  7m  20
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7)
4
7v  2  12  7
Page 97
8)
x2 7  x9
9)
y 1  3  4  y
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Review 2.3-2.5
1) Solve by completing the square:
a. x 2  8 x  11  0
b. 4 x 2  20 x  13  0
2) Caedon Hsieh lives to work in his garden. Suppose he wants to expand a 24ft by 16ft rectangular
garden by adding a border around the outer edge of the
garden. The border will be the same width
around the entire garden and the vegetables he bought will fill an area of 276 square feet. How wide
should the border be?
Simplify.
4) 3  5i 2  4i  (6  i 3 )
3) i 3045
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Solve for ALL solutions for the following equations. Be sure to check your answers.
5)
7)
2 x 2  6 x  7
27 x3  1
2
3
9) 2( x  2)  50
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6) x 4  13x 2  40  0
8) 2 x  5  3x  4
10)
x 3 5  x
Page 101
11)
12)
x  1  2 x  5x  3
x 4  4 x 2  45  0
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Answers:
1.
b.
2.
3.
4.
5.
a. x  4  5
5
x  3
2
x = width of border
i
2  5i
3  i 5
x
2
6.
x = 3 ft.
x  2i 2; x  i 5
1
1  i 3
7. x  ; x 
3
6
8. x  1
9. x = 127
10. x  7
11. x  1 , -1

x  1  2x
 
2
5x  3

2
x  1  2 2 x ( x  1)  2 x  5 x  3
2 2 x ( x  1)  3x  1  5 x  3

2 x ( x  1)

2
 2 x 1 2 


2


2
2x  2x  x2  2x  1
2
x2  1  0
0  ( x  1)( x  1)
x  1 or 1 {check solutions:


 
0  ( 2) 
 
1  1  2( 1) 

5( 1)  3

2 true statement.
1
5
12. x   i; x  
3
5
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(2.6): Inequalities
Find and graph the solutions. Use a number line and the coordinate plane.
1) x > 5
2) y < -2
Number line:
Coordinate plane:
-10 -8
-6
-4
10
10
8
8
6
6
4
4
2
2
-2
2
4
6
8
10
-10 -8 -6 -4 -2
-2
-4
-4
-6
-6
-8
-8
-10
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4
6
8 10
-10
3) 1 < x < 6
-10 -8
2
-2
4) 6 < x < 1
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Unit 4
10
10
8
8
6
6
4
4
2
2
-2
2
4
6
8
10
-10 -8
-6
-4
-2
2
-2
-2
-4
-4
-6
-6
-8
-8
-10
-10
4
6
8
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10
Solve. Graph solution on a number line, and express the solution as an interval.
5) 12 > x + 13
7) 1 
9)
x4
2
5
3
0
2m  5
6) -4 < y + 6 < 5
8) 5 – x > 11
1
10) 2  3  k  5
4
11) (x-3)(x+3) < (x+5)2
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Solve. Express answers in interval notation.
1) x – 8 > 5x + 3
2) 2 
3)
4)
2x(6x+5) < (3x-2)(4x+1)
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4x 1
0
3
3
0
2 x
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Introduction to Absolute Values with inequalities:
1) | x | < 5
2) | x + 6 | < 4
Interval
Interval
Coordinate plane:
Coordinate plane:
10
10
8
8
6
6
4
4
2
2
-10 -8 -6 -4 -2
2
4
6
8 10
-10 -8 -6 -4 -2
2
-2
-2
-4
-4
-6
-6
-8
-8
-10
-10
4
6
8 10
Solve. Express the answer as an interval.
3) | 7 – x | < 5
4) | x | < -5
5) |y| > 5
6) | y+7 | > 5
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8) | x – 4 | < 0.03
7) | -x | > 2
9) | x-3 | - 0.3 > 0.1
10) 1 < | x | < 5
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11)
2
5
| 2x  3 |
12)
2
0
| 2x  3 |
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(2.7) Advanced Inequalities
Practice: Solve for all values of the variable.
1) x4 < x2
2)
(2  x)(3x  9)
0
(1  x)( x  1)
How would your answer to #2 change if you were asked to solve over the interval of [ -2, 3.5]?
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4) Skylar Mark has a big fish tank in which she has decided to raise salmon. She is worried
that her population will grow too large. For this particular salmon population, the
relationship between the number S of spawners and the number of offspring R that survive to
4500 S
maturity is given by the formula R =
. Under what conditions is R > S?
( S  500)
5)
The population density D ( in people/mi2) in a Chicago is related to the distance
5000 x
x from the center of the city by D= 2
. In what areas of the Chicago
( x  36)
does the population density exceed 400 people/mi2?
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Review Problems from the textbook:
You can go back and re-do any problems previously assigned.
The following list is optional, and yours to do if you want more practice.
I tried to pick a variety of problems, and you should NOT make the assumption that this
that will be on the test
is all
p. 117 # 11, 15, 18, 21, 23, 25, 33, 36, 38, 40, 42, 47, 50, 53, 55, 59, 60, 62,64, 65,70, 73
Group Discussion Review:
1. When we factor the sum of cubes, x3  y 3 , is the factor ( x 2  xy  y 2 ) ever factorable over the real
numbers? Solve the equation x 2  xy  y 2  0 for x to justify your answer.
2.
Error Analysis. Hardar is solving the inequality 2 x 2  x  3 . She factors to arrive at the
step ( x  1)(2 x  3)  0 and then proceeds to:
x  1  0 or 2 x  3  0
Explain why this step is incorrect.
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