Math 90 – Cumulative Review Sheet
Chapter 2
1. Solve and graph the inequality: x > -3 and x < 7
2. Solve and graph the inequality: x > -3 or x < 7
3. Solve and graph the inequality: - 4 < 3x + 5 < 10
4. Solve: |3x – 7| = 5
5. Solve and graph: |2x – 1| < 11
6. Solve and graph: |4x + 2| > 10
Chapter 3
1. Solve: x3 – 4x2 – 45x = 0
2. Factor: 3a2 + 11a – 4
3. Factor: 8x6 – 27y3
Chapter 4
7x  2
is undefined.
x  2x 2 - 15x
y-x
Write in lowest terms: 2
x - y2
1. Find the values where
2.
3.
4.
5.
6.
7.
8.
3
x 2 - 10x  24 x 2 - 9x  18

Divide:
2x 2 - 5x - 12
4x 2 - 9
Subtract and write in lowest terms:
b
2
 2
2
b  2b - 8 b  3b - 4
3a  1 a  7

Add and write in lowest terms:
a-3
a-3
Simplify:
2
6a 1
1
3a 1
Divide: (3x 3  5x 2 - 6x - 2)  (x  4)
Divide: (x 3 - 7x - 6)  (x  2)
x
6
3 
x-6
x6
3
5
10. Solve:

2p - 1 3p  2
b
2
3

 2
11. Solve:
b - 5 b - 6 b - 11b  30
rt
12. Solve the formula A  P(1  ) for t.
m
13. A boat goes 20 miles upstream in the same time it takes
to go 60 miles downstream. The current flows at 4 mph.
Find the boat speed (with no current).
14. Joy can paint a house in 20 hours, and Amy can paint
one in 30. They work together for a while, but Amy
sprains her wrist, so Joy takes 5 hours to finish the rest by
herself. How long did they work together?
9.
Solve:
Chapter 5
3
24x 6 y 7 z 8
1.
Simplify:
2.
Simplify: 2 48  4 75  7 3
3.
Solve:
x 2  5x  20  2
Chapter 6
3  2i
4i
Use the discriminant to determine the number and type
of roots for the equation 25x2 – 11x – 4 = 0.
Solve: (3x – 4)2 = 32
Solve: -11x2 + 4x – 2 = 0
3
n5

Solve:
n2
4
Solve: 2 p  p  3
Solve and graph: y2 – 4y – 21 < 0
1. Simplify and express in standard form:
2.
3.
4.
5.
6.
7.
2x  1
4
x 1
9. To get from Modesto's city center to Turlock's center, I
can take Yosemite Blvd 6.8 miles east, then Albers/Gear
Rd 8.0 miles south to Turlock. Assuming that Highway 99
is a perfect diagonal (and there are exits at the city
centers), how many miles is the diagonal shortcut on
Highway 99 between Modesto and Turlock?
10. Find 2 consecutive even integers whose sum of squares
is 164.
11. Alex bought a number of shares of stock for $2800. A
month later, the value of the stock had increased $6 per
share, and he sold all but 60 shares to regain his original
investment of $2800. How many shares were sold?
8.
Solve and graph:
Chapter 7
1. For each of the following equation, what type(s) of
symmetry does it have?
a) y = x2 – 3x + 4
b) x2 = 4y
c) xy3 = 6
d) x2 + y2 = 9
2. Graph the equation: y = -3x + 6
-4
3. Graph the equation: y  2
x
4. Graph the equation: y = - x3
5. Graph the equation: 3y = x
6. Graph the equation: x = -5
7. A case of 24 sodas costs $4, and each soda is sold for
$0.50. If the profit is described by the equation P = 0.5x –
4, where x is the number of sodas sold,
a) Find the profit for selling 10 sodas.
b) What is the break-even point?
c) Graph the equation, putting P on the y-axis.
8. Graph the linear inequality 2x – 5y < - 10.
9. Graph the linear inequality described the equations x < -1
and y > 2.
10. Find the slope of the line between the points (-3, 4) and
(0, -1).
11. Find the distance between the points (-3, 4) and (0, -1).
12. If a symmetrical roof has a 35% pitch, and the horizontal
beam spanning the base of the entire roof is 60 ft. long,
how long is the vertical beam connecting the base and the
peak of the roof?
13. Find the equation of the line with slope = ½ which
contains the point (4, -6).
14. Find the equation of the line passing through the points (3, 4) and (0,-1).
15. Find the equation of the line perpendicular to the line 2x –
y = 7 which contains the point (-1, 3).
Chapter 8
1. Find the equation of the function, f(x), perpendicular to the
function
g(x) = -2x + 11 which contains the point (-1, 3).
3x  2, x  0
2. If f(x)  
find f(-2), f(0) and f(3)
5x  1, x  0
f(a  h) - f(a)
3. For f(x) = x2 – 3x + 7, find
h
4. For each of the following graphs, determine whether it’s a
function.
5. Find the domain and range of f(x) = x2 – 4
6. Find the domain and range of f(x)  9  x
7. The cost of producing a book is a 2000 flat rate (for print
setup), plus $6 per book for the material costs (paper and
ink).
a) Write a function describing cost, C(x), where x is the
number of books produced.
b) How much does it cost to produce 100 books?
c) If each book is sold for $10, write a function describing
revenue (money taken in), R(x), where x is the
number of books sold.
d) How much revenue is earned by selling 100 books?
e) If profit, P(x) = R(x) – C(x), what is the break-even
point?
8. Graph: f(x) = -x – 5
9. Graph: f(x) = - (x – 4)2 – 1
10. Graph: f(x) = |- x – 3|
11. Graph: f(x) = ½(x + 1)3 + 2
12. Graph: f(x) = x2 + 6x + 8
 1,...x  2

13. Graph: f(x)  1,...  2  x  0
x,...x  0

14. The height of a ball thrown straight up at an initial speed
of 96 feet/sec. Is described by the function f(x) = 96x –
16x2.
Find the highest point reached by the ball.
15. For the functions f(x) = x2 - 8x + 15 and g(x) = x – 3, find
(f  g) (x), and (f/g)(x)
(f+g)(x), (f-g)(x),
2
16. For f(x) = x – 1 and g(x) = 2x – 5, find
(f  g)(x) and (g  f)(x)
17. If f(x) = - x2 – x + 4, and g(x) = x  2 , find
(f  g)(6) and (g  f)(-2)
18. The number of hours required to paint motel rooms varies
directly as the number of rooms and inversely as the
number of people working. If it takes 3 people 10 hours to
paint 20 rooms, how many hours will it take 4 people to
paint 40 rooms?
19. If d varies jointly as m and the square root of t, and d =
24 for m = 3 and t = 4, write the formula showing the
relationship of d, m, and t.
Chapter 10
1. Solve: 94x – 2 = 1/81
2. Solve: (1/27)3x = 32x -1
3. Graph: f(x) = - 3-x
4. Is the function f(x) = -|x| - 2 one-to-one?
5. Are the functions f(x) = (x – 4)2 for x > 4 and g(x) =
x  4 for x > - 4 inverse functions?
x
6. Find the inverse of f(x) =
for x > -1
x 1
7. Find the inverse of f(x) = -3x – 6
8. Find the interval(s) on which the function f(x) = 2 x - 3
is increasing and decreasing.
9. The half-life of radium is about 1600 years. If the original
amount of radium is 500 grams, how much will remain
after 800 years?
10. Is the function f(x) = -|x| - 2 one-to-one?
11. How much will a $500 investment be worth in 10 years if
it is invested at 8% interest and compounded
continuously?
12. Simplify a) log381 b) log497 c)log10 .0001
13. Solve: log8x = -2/3
14. Express as the sum or difference of simpler logarithms
x3 z
with no exponents: log 5 ( 5 )
y
15. Express as a single logarithm:
1
log 2 s  log 2 t - 4log 2r - log 2p 3
2
16. Solve: log 3 x  log 3 (x  2)  1
17. Solve for x to 4 decimal places
a) log x = 3.9335
b) ln x = .5240
18. Graph the function f(x) = - log2x
19. Solve: 2ex = 12.4
20. Solve: 22x + 1 = 3x+1
21. An earthquake in Mexico City in 1985 had an intensity
level about 125,000 times the reference intensity. Find
the Richter number for that earthquake.
22. The quantity of bacteria in a culture after t hours is
described by the equation Q = Qo e.29t, where Qo is the
original number of bacteria. How long will it take for 100
bacteria to increase to 400 bacteria?
Chapter 11
1. Solve the system using the method of your choice:
8x + 3y = 24
-2x + y = -6
2. Solve the system using Cramer’s rule:
5x + 4y = 10
3x + 5y = 2
3. Solve the system:
y–z=4
x + y – 3z = 1
-x + 2y
=0
4. For the following word problem, write a system of
equations. Represent this system as an augmented
matrix, but DO NOT SOLVE!
“A trail mix is made from peanuts ($2/lb), raisins
($2.50/lb), and M & M’s ($4/lb). A mixture of 22 lb is
made, which costs $55 all together. If the weight of
peanuts is twice the weight of raisins, how much of each
type is used?”
5.
6.
7.
8.
A system has been reduced so that its augmented matrix
is:
1 - 3 4
1 


Solve for x, y, and z.
-4 
0 1 - 2


3 
0 0 1
Find
Find
3
6
-2 1
5
0
-3 1
1 2
-1 4 2
Solve the system:
x – 3y + z = 4
2x - 5y + 5z = -2
-3x + 9y – 3z = -12
Chapter 13
1. For the parabola y2 – 12x + 4y – 8 = 0
a. Which direction does it face?
b. Find the vertex and focus
2. Find the focus of the parabola x2 = - 12y
3. Find the equation of the circle with center at (2, -3) and
radius = 5. Write your answer in general form.
4. Find the center and lengths of axes major & minor axes
of the ellipse with equation: 9x2 – 36x + 4y2 + 16y + 16 =
0. Sketch the graph.
5. Find the equation of the ellipse with vertices at (0, 6) &
(0, -6), and foci at (0, 27 ) & (0, - 27 )
6. Find the vertices, foci, and equations of the asymptotes,
and sketch the hyperbola: y2 – 4y – 4x2 – 24x – 36 = 0
7. Find the equation of the hyperbola with vertices at (0,
 3 ), length of conjugate axis is 4.
8.
9.
Solve the system:
x2 – y2 = 4
x2 + y2 = 4
Graph the system in # 8, showing point(s) of intersection.
Chapter 14
1. Find the general (nth) term of the sequence: 5, 8, 11,
14…..
2. Find the sum of the first 50 terms of the sequence 2, 6,
10, 14……
25
3.
Find
 3i  2
i1
Find the 10th term of the sequence 1, -2, 4, -8……
Find the sum of the first 10 terms of the geometric
sequence 5, 10, 20, 40…
6. If a child receives a penny on Feb. 1 and each day,
doubles the amount of money received, how much will he
receive on Feb. 28?
7. A pile of logs has 30 logs in the bottom layer, 29 in the
next layer, 28 in the 3rd layer, with one less in each
successive layer, and 1 log in the top layer. How many
logs are in the pile?
8. A tank holds 1024 gallons of water. Each week, ¼ of the
tank evaporates. How many gallons of water remain after
5 weeks?
9. Find the sum of the infinite geometric sequence,

1
5  ( )i

3
i 1
10. Change the repeating decimal .321321… to a reduced
fraction
4.
5.
Chapter 15
1. If a restaurant has 4 salads, 9 entrees, and 5 desserts,
how many different meals with a salad, an entrée, and a
dessert can be arranged?
2.
In how many different ways can 4 letters be dropped into
6 mailboxes?
3. In how many different ways can 4 letters be dropped into
6 mailboxes if no mailbox has more than 1 letter?
4. How many committees of 2 Republicans and 2
Democrats can be chosen from a group of 5 Republicans
and 7 Democrats?
5. How many 4-person committees of at least 1 man can
be formed from a group of 5 women and 3 men?
6. How many leadership teams of a president, a vicepresident, and a secretary can be selected from 10
people?
7. How many ways can A, B, C, D, E, and F be arranged in
a line so that A and B are standing next to each other?
8. How many 5-card hands with 3 aces and a pair of face
cards can be dealt from a deck of 52 playing cards?
9. Expand the binomial (a + 2b)6
10. Find the 5th term of (3x – y)7