Download Algebra Semester 1 Review - Rochester Community Schools

Algebra Semester 1 Review
Name___________________________________
In order to earn full credit, ALL WORK must be shown on ALL problems.
The questions that are circled are those that you should not use a calculator to solve.
1.)
To find the mileage, or how many miles per gallon a car can travel, you can use the expression
m
where m is the distance in miles and g is the number of gallons of gas used. Find the mileage
g
for a car that travels 350 miles on 14 gallons of gas.
2.)
Evaluate the expression n  3  27  3 , given n = 3.
3.)
Simplify 53  62  53  32  3  2
4.)
3
Evaluate the expression 16  12 x  x , when x = 3.
5.)
Simplify 7  62  7  32  4  3




a.) 27
6.)
3
b.) 243
c.) 189
d.) 261
Suppose one person can do a job in x minutes, and a different person can do the same job in y minutes. The
time (in minutes) in which the job can be completed by both people working together can be found using the
expression
xy
x y
. If it takes you 6 minutes to sweep your kitchen floor and it takes your brother 8 minutes,
how long would it take both you and your brother working together?
7.)
It is known that a cyclist can travel 41.4 miles in 3 hours. At that rate, how far can the same
cyclist travel in 7 hours?
a.) 95.2 miles
b.) 97.8 miles
c.) 96.6 miles
d.) 97.4 miles
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8.)
A shipping service charges $0.43 for the first ounce and $0.29 for each additional ounce of package weight.
Write an equation to represent the price P of shipping a package that weighs x ounces, for any whole number
of ounces greater than or equal to 1.
9.)
A store that sells gift baskets is having a promotional sale. Customers can make their own fruit baskets to use
as gifts. Customers pay $3.00 for a basket and add $0.20 per pound for all types of fruit. The cost for a
basket containing p pounds of fruit is $4.30. Which equation could be used to find p, the number of pounds of
fruit in this basket?
a.)
3.00 + 0.20p = 4.30
b.)
(0.20 + 4.30)p = 3.00
c.)
3.00(4.30 + p) = 3.00
d.)
0.20 + 3.00p = 4.30
10.)
Write an equation equivalent to the verbal statement: "three times the sum of a number n and 7 is 16."
11.)
Make an input-output table for the function y = 2x + 4. Use x -values of 1, 2, 3, 4, and 5.
X
y
12.)
Does the following data represent wind speed as a function of lift?
Wind speed (mi/h)
Lift
13.)
(ft/s)
20
30
40
4.6
22
40
32
Which of the functions represents the input-output table?
Input
0
1
2
3
14.)
10
Output
3
5
7
9
Functions
y = 2x – 3
y = 2x + 3
y = 2x – 4
y = 3x + 3
Find the range of the function.
Input
1
9
4
Output
11
6
5
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2
15.)
List a value of x so that the relation is not a function. {(6, 2), (5, 3), (1, 0), (x , 4)}
16.)
Evaluate the expression   12 .
17.)
Use the concept of opposites to simplify −(– (–(–(4)))).
18.)
Heather is selling fruit as a fund raiser for the soccer team. She has sold 5 pounds so far.
The graph shows the profit related to the amount of fruit sold.
19.)
a.)
Write a function rule for the graph. Use the function rule to find the
profit if Heather sells a total of 30 pounds of fruit.
b.)
Another student claims that Heather must sell 5 pounds of fruit to make
a profit of $2. Is this true? Why or why not?
Find the sum.
a.) –4 + (–13)
20.)
b.) 28.43 + (–13.17) + 2.07.
Melissa keeps track of the profit made each month by her business. Her profits for the three
months of June, July, and August were $26,985, –$13,400, and –$4,042.
a.)
What was her total profit for the summer?
b.)
Melissa adds the last two amounts in the list first, then adds that sum to the first amount.
Her business partner adds the first two amounts first, then adds that sum to the last amount.
Will they get the same sum? Why or why not?
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3
21.)
Evaluate –(–4) – (–5) + 6
22.)
Find the change in temperature from –13°C to 15°C.
23.)
Simplify the expression (–7) + 6 + [–(2 – 3)].
24.)
Evaluate the expression 28   x   10 when x = –15.
25.)
Simplify
26.)
Simplify the expression
27.)
Simplify: 5(9x – 5y)
28.)
Bill wants to simplify the following expression: 5(3x – 2y) + 2(x + 2y) – 3(3x – 2y)
Which of the following expressions is equivalent to the expression above?
(–7)(5)(6)
a.) 8x
10(2c )
b.) 8x – 12y
c.) 8xy
d.) 8x – 8y
29.)
Simplify: –4(x + 3)
30.)
Simplify the expression –2(2x + 8(–3 + x))
31.)
Pat wrote this mathematical sentence: 9x 3  3x 3  12 x 6 . Explain her mistake and how to correct it.
32.)
 4
Find the quotient. 12    
 9
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4
33.)
What is the multiplicative inverse of
a.) 19
34.)
35.)
b.) 9
Find the quotient 8  2
a.) 3
9
?
19
19
9
d.)
10
19
c.)
2
3
d.)
3
64
2
3
b.) 21
1
3
Which of the following is an irrational number?
a.)  14
b.)
5
6
c.) 0.575757...
36.)
Find all square roots of the number 0.64.
37.)
Solve the following equations:
a.) x  12  26
38.)
c.)
b.) 
x
5
d.)  9
 15
c.)
2
x  30
3
A college student has set aside $240 for the rest of the school year to use the coin-operated laundry facility in
his dormitory. Each time he uses the machines, it costs $7.50. Choose the equation that represents the
amount remaining in his fund, f, after he has done laundry x times. Find the amount remaining in the fund
after 12 trips to the laundry facility.
a.) f = 240 – 7.50x ; $90.00
b.) f = 240 – 7.50x ; $150.00
c.) f = 7.50 – 240x ; $150.00
d.) f = 7.50 – 240x ; $90.00
39.)
The temperature was x° F. It rose 15° F and is now 39° F.
Write and solve an equation to find the original temperature.
40.)
Gary's cellular phone bill averages $56.35 per month, based on a fixed fee of $29.95 and $0.22 per minute of
usage. The phone company has decided to reduce the per-minute charge to $0.18. How much does this
change save Gary in an average month?
41.)
You are going to varnish a floor that is 27 ft. by 63 ft. One pint of varnish covers 23 yd2 of space.
How many pints of varnish do you need?
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5
42.)
John works as a salesman for a cellular telephone service. He is paid $250 per week plus $15 for each
contract a customer signs. He needs to earn at least $450 per week in order to meet his living expenses.
What is the fewest number of contracts he needs to sell each week to meet his living expenses?
43.)
Michelle wants to earn $900 selling 22 knit scarves. She wants to sell each scarf for $4 less than her
competitor. If x is the price charged by her competitor, which equation models the situation?
a.) 2(22) + 2x = 900
b.) 22x = 900
c.) 22(x – 4) = 900
d.) 22 + 4x = 900
44.)
A rental car agency charges $14 per day plus $0.14 per mile to rent a certain car. Another agency charges
$18 per day plus $0.07 per mile to rent the same type of car. How many miles per day will have to be driven
for the cost of a car from the first agency to equal the cost of a car from the second agency?
45.)
Solve the equation.
a.)
4
y  28  0
10
b.)
x
2

x
4
5
d.)
25x
 7x  12
5
e.)
g.)
3  4z  5  8z
h.) 3x  3  x  4
j.)
80
5

4
w
k.)
1
y  3  7
4
3
x 4

c.) 7n  23  5n  59
f.) 5n  2n  2  11
i.) 5x  14  2x  9  4x  2
5
x
page
6
46.)
Two machines can complete 5 tasks every 4 days. Let t represent the number of tasks these machines can
complete in a 31-day month. Which proportion can you use to find the value of t?
a.)
31
t

10 41
b.)
4
t

31 5
c.)
5
t

4 31
d.)
4
t

5 31
47.)
A cyclist can travel 29.6 miles in 2 hours. At this rate, how far can the same cyclist travel in 45 minutes?
48.)
Madison solved the following equation:
2
2x  5  6 .
5
As a first step, she rewrote the equation as 2 x  5  15 .
a.)
By what number did Madison multiply both sides of the equation? Why did she choose this number?
b.)
What is the solution of the equation?
49.)
Raul uses 6.97 pounds of ground beef to make a casserole that serves 17 people. How many pounds would
he need to make the casserole for 13 people?
50.)
A classroom contains 18 girls and 24 boys. Write the ratio of girls to boys in the classroom as a fraction in
lowest terms.
51.)
Of every 5 hot dogs Martha sold, 3 had sauerkraut. What percent of the hot dogs sold had sauerkraut?
3
%
a.) 6%
b.)
c.) 60 %
d.) 0.6%
5
52.)
Shown below is a picture and its enlargement. The width and length of the smaller picture are 9 and 12
inches respectively. The length of the larger picture is 28 inches. What is the width of the larger picture?
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7
53.)
A survey was taken to determine the favorite subjects of 8th graders. The results are shown in the graph.
About what percent of students chose Social Studies as their favorite subject?
54.)
Solve y =
55.)
Graph the equation x = –5.
5
b + 10 for b.
8
56.)
What is the range of the function in the graph?
y
70
60
50
40
30
20
10
0
57.)
Sketch the graphs of x = –2 and y = –4.
Find the point at which the two graphs intersect.
58.)
State the x - and y -intercepts of the line
with the equation y = –2x + 4.
59.)
Sketch the line given by, 3x – 4y = –12.
Label the x - and y -intercepts.
1
2
3
4
5
6
7 x
page
8
60.)
Find the slope of the line that contains (–8, 2) and (7, –4).
61.)
Find the slope of the line passing through the points A (8, 1) and B (–3, –6).
62.)
Explain the difference between positive slope and negative slope.
63.)
Rewrite the equations in slope-intercept form and graph the line.
a.) 5x – 2y = 7
b.) 3x – y – 2 = 0
1
x  6 . Is the line parallel to y  6x  6 ?
6
64.)
Find the slope and y-intercept of the line y  
65.)
The depth of the water in Jeanne's hot tub varies directly with the number of minutes that the faucet is
turned on. At 8:15 A.M. Jeanne started filling her circular hot tub with water. At 8:59 A.M. there was a foot of
3
water in the tub. When will the hot tub have 2
feet of water in it?
4
a.) 11:15 A.M.
b.) 10:59 A.M.
c.) 10:15 A.M.
d.) 10:16 A.M.
page
9
66.)
A monthly phone bill, b(m), consists of a $28 service fee plus $0.13 per
minute, m, of long distance calls, given by the function b(m) = 28 + 0.13m.
Draw a graph for up to and including 120 minutes of long distance
calls made in a month. Estimate the bill if 84 minutes of long distance
calls are made.
3
x.
4
67.)
Graph the function g( x ) 
68.)
Find the value of x so that f(x) = 13. f(x) = x – 10
69.)
Choose an equation, in slope-intercept form, of a line with a slope 7 and a y-intercept of –9.
a.) y = 7x – 9
70.)
b.) y = 7x + 9
c.) x = 7y – 9
d.) y 
1
x 9
7
When Aidan had his picture taken, the photographer charged
a $10 sitting fee and $6 for each sheet of pictures purchased.
a.)
Write the function for the situation, where x is the
number of sheets purchased.
b.)
Graph this function.
page 10
71.)
Write an equation in slope-intercept form of the graph.
•
•
72.)
The cost of a school banquet is $95 plus $15 for each person attending. Write an equation that gives total
cost as a function of the number of people attending. What is the cost for 77 people?
a.)
y = 16x – 95 ; $1060
b.)
y = 95x + 15 ; $7330
c.)
y = 15x + 95 ; $1250
d.)
y = 95x + 15 ; $7300
1
and y-intercept –4.
3
73.)
Write an equation of the line with slope
74.)
Write an equation of the line shown.
75.)
Write an equation for the function in the form, f (x) = mx + b.
f (–4) = –33, f (0) = –5
76.)
Write an equation of the line passing through the point (–7, –6) with slope m = 4.
77.)
The graph for a stable that charges a $20 flat fee plus $10 per hour for
horseback riding is shown. How will the graph change if the stable changes
its charges to a flat fee of $30 plus $15 per hour?
page 11
78.)
Write the equation in slope-intercept form of the line that passes through the points (7, –1) and (2, 9).
79.)
Write an equation, in point-slope form, of the line that passes through the point (6, –5) and has the slope
80.)
Write an equation in point-slope form of the line that passes through the points (–5, –4) and (6, 3).
a.)
c.)
y 4
y 5 
7
(x  5)
11
7
( x  4)
11
b.)
y 4
11
(x  5)
7
d.)
y 5 
11
( x  4)
7
1
2
1
.
2
81.)
Write an equation in point-slope form of the line that passes through (–7,1) and has slope m =
82.)
Write an equation in point-slope form of the line that passes through the points (–2, –5) and (–6, 4).
83.)
Write the equation of the horizontal line passing through the point (7, 4).
84.)
Write the equation of the vertical line passing through the point (–5, 2).
85.)
Write an equation of the line with undefined slope that passes through the point (8, –1).
86.)
The clearing house has resistors that sell for $3.50 each and circuit boards that sell for $2.25 each.
Write an inequality that represents how many of each type of electronic equipment can be bought with at
least $7.
87.)
Write an equation in standard form of the line that passes through the two given points. (–3, 2) and (1, – 4).
page 12
88.)
Write an equation of the line that passes through (–18, –4) and is parallel to the line y 
89.)
Which of the following lines is NOT parallel to the line shown in the graph?
a.)
3x  y  3
b.) y  3x  9
c.)
12x  4y  9
d.) 3x  y  3
1
x 3.
3
•
•
90.)
Write an equation of the line that passes through (4, 0) and is perpendicular to the line y 
91.)
Tell whether x and y show a positive correlation, negative correlation,
or relatively no correlation.
92.)
Bianca is making home-made cards to send to friends
1
x 1.
5
and family and to sell at the local craft fair. This scatter
plot shows the total number of cards she had made after
each hour she worked on the task. Using this information,
predict the number of cards that Bianca can make in 12 hours.
93.)
A biology student recorded data for a science project and graphed
them on a scatter plot.
a.)
Draw a line that appears to fit the points in the scatter plot closely.
b.)
Write an equation for the line you drew.
Explain how you wrote the equation.
c.)
Use the equation you wrote in part b to approximate the
value of y for x = 1.5
page 13
94.)
Solve and graph the inequality: –1.4x < –11.2
95.)
Solve and graph the inequality: y + 9 ≥ 0
96.)
Solve and graph the inequality: 4x + 4 ≥ 2(x – 2)
97.)
Your veterinarian tells you that a healthy weight for your dog
is between 70 and 80 pounds. Write an inequality to
represent your dog's healthy weight w in kilograms.
98.)
Solve and graph the inequality: –8 ≤ 4x + 20 ≤ 8
99.)
Solve and graph the inequality: x + 3 ≤ 4 and –2x < 10
100.) Solve:
4x  2  3
101.) Solve and graph the inequality: │4x – 2│< 3
102.)
Solve and graph the inequality x  3  3
page 14
103.)
Graph the inequalities.
b.) y < 3 x − 2
a.) x ≤ –7
7
x  y  0
104.) Solve the system 
1
y  3  x
2

by graphing.
5x  2y  32
105.) Use the addition method to solve the system of equations. 
x  4  2y
106.) Mr. Frankel bought 7 tickets to a puppet show and spent $43. He bought a combination of child tickets for
$4 each and adult tickets for $9 each. Which system of equations below will determine the number of adult
tickets “a” and the number of child tickets “c” he bought?
a  c  9
9a  4c  43
a.) 
9a  4c  43
a  c  7
b.) 
107.) Solve the system for the value of x.
a  c  7
a  c  301
c.) 
4a  4c  50
a  c  7
d.) 
x  y  0

2x  y  9
108.) Solve the linear system by any method.
page 15
2

y  x  2
3
y   x  3

x  y  1
x  y  3
a.) 
b.) 
109.) The length of a rectangle is 8 cm more than four times the width. If the perimeter
of the rectangle is 46 cm, what are the dimensions?
x  4 y  1

2x  y  7
110.) Use the substitution method to solve the system of equations.
111.) A rental car agency charges $15 per day plus 11 cents per mile to rent a certain car. Another agency charges
$18 per day plus 8 cents per mile to rent the same car. How many miles will have to be driven for the cost of
a car from the first agency to equal the cost of a car from the second agency? Express the problem as a
system of linear equations and solve using the method of your choice.
112.) The sum of the ages of Petra and her mother is 53. Her mother is 11 years more than twice as old as Petra.
How old are Petra and her mother?
3x  6 y  9
113.) Solve the system of equations using the elimination method: 
x  6 y  11
114.) Which system of equations has no solution?
7x  9y  5
a.) 
 21x  27y  14
7x  9y  5
b.) 
4 x  7y  14
7x  9y  10
c.) 
7x  36y  40
7x  9y  5
d.) 
14 x  19y  10
page 16
6x  4 y  10
115.) Which of the following describes the solution(s) of the system. 
8x  12y  20
a.) No Solution
b.) (–7, 13)
c.) (5, –5)
d.) (–1, 4)
24x  8y  24
116.) Which choice best describes the solution(s) of the system of equations? 
 15x  5y  15
a.)
Infinitely Many Solutions
b.)
(–1, 0) is the only solution
c.)
(1, 48) is the only solution
d.)
No Solution
Solve the system of linear inequalities by graphing.
117.)
y  2 x  1

y  3
118.)
y  x  4

y  2x  8
page 17