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Final Exam Review
HOMEWORK: Review all in class review problems,
cumulative review problems below, ALL exams, quizzes and
homework problems, ALL Story Problem Applications from
sec 1.5, 2.4, 3.4 & 5.6, Solving equations algebraically (linear,
quadratic, rational, radical, literal, polynomial, linear systems
of equations etc) and terms and definitions.
Cum.
Review
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Cum.
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Cum
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Cum.
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619 - 620
1, 2, 9, 11, 13 – 17, 20, 22 – 25
578 – 579
1 – 9, 12 - 16, 18, 19, 21 – 28, 30
513 – 514
1 – 7, 10 – 20, 22 – 26, 28
422 – 423
1 – 5, 7, 8, 10, 12 – 19
358 – 359
1 – 11, 13 – 17, 19 – 25
245
1 – 15
203 - 204
1 – 20
Page 1
Final Exam Review
From Exam 4:
Quadratics
1. Solve 2 x  4 x  7 algebraically. Give solution in exact
simplifies form. Show all your work.
2
Must first get 0 on one side
2x  4x  7  0
2
a =2, b = -4, c = -7
 (4)  (4)  4  (2)(7)
x
22
2
4  16  56
x
22
Must put in ( ) around b!
Watch negative signs!
(-#)^2 is always positive!
x
4  72
22
x
4  36  2
22
x
46 2
22
Simplify under radical
Must reduce!
Factor and cancel like factors!
Page 2
Final Exam Review
x
2  2  3 2 
22
Factor out the 2. Then cancel.
x
23 2
2
Note: You CANNOT cancel the 2.
You can NEVER cancel over addition
and subtraction! Not equal to x  3 2 !
2.
Solve algebraically. Show all your work.
x  1  5  x
x 1  5  x
x  1   5  x
2
Must isolate radical first!

2
( x  1)( x  1)  5  x
x  2x  1  5  x
2
Square both sides.
NEVER square each term!
*** MUST FOIL!!! *****
x  3x  4  0
Get 0 on one side
( x  4)( x  1)  0
Factor!
2
x40
or x  1  0
Use zero property to set each
factor=0
Page 3
Final Exam Review
x = -4 or x = 1
Solve
MANDATORY Check in ORIGINAL EQUATION!!!
x  1  5  x
x = -4
 4  1  5  (4)
 4  1  9
 4  1  3
x=1
1  1  5  (1)
1  1  4
1  1  2
11
So answer is only x=1
Page 4
Final Exam Review
A model rocket is launched straight upwards from the top of a
100 ft balcony. The initial velocity is 160 ft/s. The height of
the rocket in feet h(t) at any time t in seconds is given by
h(t )  16t  160t  100 Find the following.
3.
2
a) Find the coordinates of the vertex of
h(t )  16t  160t  100 . Show your work!
b
160

5
x-coordinate = 
2a
2(16)
b
y-coordinate = h( )  h(5)  16  5  160  5  100  500
2a
2
2
ANSWER 5, 500 
b) How long does it take the rocket to reach its maximum
height? Include units.
ANSWER 5 sec
c) What is the maximum height the rocket will reach? Include
units.
ANSWER 500 ft
d) How long does it take for the rocket to hit the ground?
Include units. Show your work!
Hits ground when h = 0
0  16t  160t  100
2
Solve with QE
Page 5
Final Exam Review
 160  160  4  (16) 100
t
2  16
2
ANSWER 10.59 sec (Round answer to 2 decimal places)
4. Rationalize the denominator and simplify the expression, if
possible. Show all your work.
5
7 3
1
5. Solve 4 x  7  3  2 x or x  3  2 x  6 algebraically.
2
Show all work. Give final answer in interval notation.
4x  7  3  2x
1
x  3  2x  6
2
2 x  10
x  6  4 x  12
x5
 6  3x
2 x
Page 6
Final Exam Review
MUST graph on number line (important part of work)
1
4 x  7  3  2 x or x  3  2 x  6
2
Solve 4 x  7  3  2 x and
Solve 4 x  7  3  2 x or
1
x  3  2x  6
2
1
x  3  2x  6
2
Solve 4 x  7  3  2 x and
1
x  3  2x  6
2
Page 7
Final Exam Review
Simplify completely. Assume all variables represent positive
real numbers. Do not leave negative exponents in your
answer. Show all your work.
 8x y

 27 x z
8
1
 8x y

 27 x z
2y

3x z
8
1
3
4
5
1
4
3
 53
3
4
5



1
3
1
3
1
3
8 (y )
(2 ) y
  8y 

 
 
(3 ) x z
  27 x z  27 ( x ) ( z )
2y z

3x
3
4
9
5
1
1
3
3
3
9
1
4
3
1
3 1

4 3
3 13
3
5
1
3
3 13
9 13
5 13
5
3
1
4
3
Note: WE NEVER convert to radical form unless instructed to
do so. When the problem is given in exponential form, keep it
in this form!
ALWAYS SHOW ALL STEPS! EVERY STEP ABOVE is
required!
Multiply and simplify. Show all your work.
(3 10  2 )(4 5  5 8 )
MUST FOIL!!! Reduce ALL radicals!
Page 8
Final Exam Review
Combine. Assume that all variables represent positive
numbers. Show all your work.
6 50a b  4ab 18ab  ab
3
5
3
2
8ab
Always reduce all radicals to like terms first. Then combine
with dist property.
6 5  2a ab b  4ab 3  2ab b  ab
2
2
6  5ab
30ab
2
2
4
2
2
2
2  2ab
2
2ab  4ab  3b 2ab  ab  2 2ab
2
2ab  12ab
2
2ab  2ab
2
2ab
30  12  2ab 2ab
2
20ab
2
2ab
Write the expression by using exponents rather than radical
3
notation.
x
12
5
Which of the following is equivalent to

5  3 ?
2
NEVER square each term. ALWAYS F.O.I.L!!!
Page 9
Final Exam Review
Rationalize the denominator.
2
5x
NEVER square an expression. ALWAYS multiply by “1” to
make denominator be powers of the index.
Solve 2( x  5)  48 algebraically. Give the answer exactly.
2
Isolate the squared term first. Never FORGET  !
Do not multiply all out!
Page 10
Final Exam Review
Equations vs expressions:
NEVER DROP = in EQUATIONS!!!!!
Linear: Solve  2  3(4 x  (5  2 x)  1)  2 (3x  1)
2
Simplify  2  3(4 x  (5  2 x)  1)
Page 11
Final Exam Review
Polynomial:
Solve. Show all your work.
12n  3n
3
Always factor completely! Watch constants!
Factor 12n  3n
3
Note: There is NO = sign, so answer is an expression! DO NOT
add an equal sign when one is not present!
Page 12
Final Exam Review
Solve
14
x
2
algebraically


2 x  5x  3 2 x  1 x  3
2
Note: When there is an equation (= sign) ALWAYS multiply
by LCD!!!
MUST CHECK ANSWER!
Page 13
Final Exam Review
Simplify
14
x
2


2 x  5x  3 2 x  1 x  3
2
Note: When there is NO = sign, NEVER NEVER NEVER
multiply by LCD!!!
You MUST put each term over the LCD by multiplying by “1”
to make each term have a denominator equal to the LCD and
them combine! NEVER DROP DENOMINATORS! Rewrite
entire fraction in EACH STEP when you simplify!
Page 14
Final Exam Review
Literal equations: Solve
3b  7au
 4 y  1 for u.
5 x  2u
Strategy: Clear Fractions by mult by LCD. Get all terms with
the variable you want on one side and terms without that
variable on the other. Factor out the variable you want. Then
divide by the coefficient.
Page 15
Final Exam Review
6 x  xy  y
6 xa  10 xb  3ay  5by
Simplify

50 x  18x 15ax  25bx  9ax  15bx
2
2
3
Answer:
2
2
3x  y
2(5 x  3)
Page 16
Final Exam Review
Solve 3x  2 y  11
2 x  5 y  18
Answer: (-1,4)
Solve 2 x  5 y  18
5
x   y9
2
5


Answer: Infinite solution  x x   y  9
2


Page 17
Final Exam Review
Domain: The set of all input values: Find the domain of the
following: Give answer in interval notation if possible.
5x  2
f ( x) 
16x  36 x
3
Answer: (,  3 )  ( 3 ,0)  (0, 3 )  ( 3 , )
2
2
2
2
g ( x)  3  5 x
Answer: (, 3 ]
5
 7,3,2,4,5, 1,8,9
Answer:
 7, 2, 5, 8














y
x







Answer:  3, 6
Page 18
Final Exam Review
Lines:
Parallel Lines have same slopes.
Perpendicular lines have opposite reciprocal slopes.
Slope = rise/run
Find the equation of the line perpendicular to 2 x  5 y  7
going through (-1, 3). Give answer in slope intercept form.
Find the equation of the line parallel 2 x  3 going through
(4, -1). Give answer in slope intercept form.
Find the equation of the line perpendicular 5 y  1 going
through (-2, -3). Give answer in slope intercept form.
Page 19
Final Exam Review
Find the equation of the line going through the points
(-3,5) and (2, -1).
Find the x and y intercepts of the line going through the
points (-3, 5) and (2, -1). Sketch the graph.
Page 20
Final Exam Review
Algebraic Properties
A) Associative Property of Addition
B) Associative Property of
Multiplication
C) Commutative Property of Addition
D) Commutative Property of
Multiplication
E) Identity Property of Addition
F) Identity Property of Multiplication
G) Inverse Property of Addition
H) Inverse Property of Multiplication
I) Distributive Property
J) Multiplication Property of 0
i)

iii)
v)
3 3
 0
5 5
_____
5  (6  7)  (5  6)  7
 3  5
    1
 5  3
vii)
ii)
____
___
2 6 2
4 4
(  ) 
3 7 5
7 15
___

3
3
0 
5
5
____
iv)
5  (6  7)  5  (7  6)
vi)
3
 3
   1  
5
 5
viii)
____
_____
2(3  5)  2(5  3) ____
Page 21
Final Exam Review
Classify the numbers by set. Check all boxes that the number belongs to
Number
Natural Whole Integers Rational Irrational Real
 16
0

-5.36
2

4
4
16
2
5
3
5

8
3 8
2.345345345 
2
5
17
List the rational #’s in the 1st column above:
List the irrational #’s in the 1st column above
Page 22
Final Exam Review
Applications:
Linear:
Example: The sum of 2 consecutive odd integers is -56.
Find the integers.
Answer: -27 & -29
Example: Suppose you plan to borrow $6000 from 2
lenders to pay for your tuition next year. One lender
charges 10% simple interest and the other charges 4%
simple interest. How much did you borrow from each
lender if you paid a total of $391.83 in interest after 1
year?
Page 23
Final Exam Review
Answer: $2530.50 from the 10% lender and $3469.50 from
the 4% lender.
Example: Suppose you invested $5000 into 2 accounts, a
CD paying 6% simple interest and a stock paying 8%
simple interest. If you receive $706.40 in interest after 2
years, how much was invested in each account?
a) Set up as a an equation in one variable and solve
b) Set it up as a system of equations and solve.
Answer: a) Let x = amt in CD. Then
0.06x*2 + (5000 – x) * .08*2 = 706.4
b) Let x = amt in CD and y = amt in stock
x + y = 5000
0.06x*2 + 0.08y*2 = 706.4 or .12x + .16y = 706.4
CD: $2340; Stock: $2660
Page 24
Final Exam Review
Example: How many liters of 40% antifreeze should be
added to 4 L of a 10% antifreeze solution to produce a 35%
antifreeze solution?
a) Set up as a an equation in one variable and solve
b) Set it up as a system of equations and solve.
Answer: a) Equation: Let x = amt of 40% antifreeze soln
0.40x + 0.10*4 = 0.35 ( 4+x)
b)Equation: Let x = amt of 40% antifreeze soln, and y
be the amt of 35% soln
x+4=y
0.40x + 0.10*4 = 0.35*y
Answer: 20 L of 40% solution
Page 25
Final Exam Review
Example: How many liters of 2.5% bleach should be added
to a 10% bleach solution to produce 600 ml a 5% bleach
solution?
a) Set up as a an equation in one variable and solve
b) Set it up as a system of equations and solve.
Answer: a) Equation: Let x = amt of 2.5% bleach soln
0.025x + (600-x)0.10 = 0.05 *600
b)Equation: Let x = amt of of 2.5% bleach soln and y
be the amt of 10% soln
x + y = 600
0.025x + 0.10y = 0.05*600
Answer: 400 ml of 2.5% and 200 ml of 10%
Page 26
Final Exam Review
Example: A favorite blend of coffee is a mixture of
Columbian costing $5.10 per pound and Hazelnut costing
$6.40 per pound. How much of each should be used to
produce 10 lbs of the blend costing $5.85/lb?
a) Set up as a an equation in one variable and solve
b) Set it up as a system of equations and solve.
Answer: a) Equation: Let x = amt of Columbian coffee
5.10x + 6.4(10-x) = 5.85*10
b)Equation: Let x = amt of Columbian coffee and y be
the amt of Hazelnut coffee
x + y = 10
5.10x + 6.4y = 0.1*5.85
Answer: 4.23 lbs Columbian and 5.77 lbs Hazelnut
Page 27
Final Exam Review
Example: Two trains are 190mi apart and travel toward each
other on the same road. They meet in 2 hours. One travels 4
mph faster than the other. What is the average speed of each
train? Set up as an equation in one variable and solve.
Answer: D=rt table. TOTAL DISTANCE (SUM)
Let x be the rate of the first train.
Equation 2x +2(x+4)=190
1st train 45.5 mph, 2nd train 49.5 mph
Page 28
Final Exam Review
Example: Bob can ride his bike 2 mph faster than Johann can
rollerblade. If it takes Bob 3.1 hours to ride his bike down a
trail and Johann 4.3 to rollerblade the same trail, how fast were
each of them going?
Answer: D=rt table. Distance same
Let x be the rate of Johann.
Equation 3.1(x+2)=4.3x
Johann 3 mph, Bob 5 mph
1
6
1
6
Page 29
Final Exam Review
Example: It takes a plane 3 hours to travel 600 miles with a
tailwind and 5 hours to return the same distance with a
headwind. Find the speed of the plane and the speed of the
wind.
Answer: D=rt table. {headwind/tailwind with same distance}
Let x be the rate of plane in still air and w = rate of
wind.
Equation 3(x+w)=600
5(x-w)=600
Plane 160 mph, wind 40 mph
Page 30
Final Exam Review
Example: A motorist travels 80 mi while driving in a bad
rainstorm. In sunny weather, the motorist drives 20mph faster
and covers 120 mi in the same amount of time. Find the speed
of the motorist in the rainstorm and in sunny weather.
Answer: D=rt table. SAME TIME
Let x be the rate in rainstorm
80
120
Equation

x x  20
In rainstorm 40 mph, Sunny Day 60 mph
Page 31
Final Exam Review
Example: Brooke walks 2 km/hr slower than her older sister
Pam. If Broke can walk 12 km in the same amount of time that
Pam can walk 18 km, find their speeds.
Answer: D=rt table. SAME time t=D/r
Let x be the rate of Pam
12
18
Equation

x2 x
Pam 6 mph, Brooke 4 mph
Page 32
Final Exam Review
Example: A bicyclist rides 60 mi against the wind and returns
60 mi with the wind. His average speed for the return trip is 2
mph faster. How fast did the cyclist ride each way if the total
time for the trip was 11 hrs?
Answer: D=rt table. Total Time
Let x be the rate of against wind
60 60
Equation

 11
x x2
10 mph against the wind, and 12 mph with the wind
Page 33
Final Exam Review
Example: A local theater charges $8.50 for student tickets and
$12 for general admission. If they sold 167 tickets for $1864,
how many of each type did they sell?
Equation: x  y  167
8.50x  12 y  1864
Answer: 40 student tickets and 127 general admission.
Example: The width of a rectangle is 5 in less than 3 times the
length. The area is 28 in2. Find the dimensions.
Equation: 28 = l(3l-5)
Answer: l=4 in, w=7in
Page 34
Final Exam Review
Example: A shadow cast by a yardstick is 2 ft long. A shadow
cast by a tree is 11 ft long. How tall is the tree?
3 h

2 11
Answer: 16.5 ft
Equation
Example: A 64 oz bottle of laundry detergent costs $4.00, how
much would a 100-oz bottle cost?
64 oz 100 oz

4$
x$
Answer: $6.25
Equation:
Page 35
Final Exam Review
Example: A chemist mixes water and alcohol in a 7 to 8 ratio.
If she makes a 450 ml solution, how much is water and how
much is alcohol.
7
x
7
x
or


8 450  x
15 450
Answer: 210 ml water and 240 ml alcohol.
Equation:
Example: The cost of a monthly texting plan is $12 per month
plus $0.10 per text.
a) Find a formula for the cost C as a function of the number
of text messages x.
b) How much will it cost you to send/receive 50 text
messages? 2500 text messages?
c) How many text messages would you need to send/receive
to make it more economical to switch over to the unlimited
plan with a fixed rate of $65 per month?
Answer: a) C  12  0.1x b) $17, $262
c) > 530
Page 36