Name ________________________________________________
Short Answer: Show all work and
leave answers in fraction form.
1. Julia
wrote 14 letters to friends each month for y months in a row. Write
an expression to show how many total letters Julia wrote.
2. Evaluate the
expression q v for q = 5 and v = 1.
3. Subtract.
5 (8)
4. Evaluate 5u
for u = 4.
5. Divide.
6. Divide.
0
Έ
5.928
7. Simplify
.
8. Simplify
.
9. Simplify
.
10. Write 9 as a
power of the base 3.
11. If the
population of an ant hill doubles every 10 days and there are currently 40
ants living in the ant hill, what will the ant hill population be in 20
days?
12. Find the square
root.
13. The area of a
square garden is 202 square feet. Estimate the side length of the garden.
14. Simplify
.
15. Simplify
.
16. Simplify the
expression
.
17. Tatia has coins
in pennies, nickels, dimes, and quarters. The total amount of money she has
in dollars can be found using the expression (P + 5N + 10D
+ 25Q)
Έ
100. Use the table to find how much money Tatia has.
18. Use the numbers
2, 3, 5, and 8 to write an expression that has a value of
.
You may use any operations, and you must use each of the numbers at least
once.
19. Simplify the
expression
.
20. Write 11 47
using the Distributive Property. Then simplify.
21. Simplify by
combining like terms.
22. A phone company
advertises a new plan in which the customer pays a fixed amount of $25 per
month for unlimited calls in the country, and $0.10 per minute for
international calls. Find a rule for the monthly payment a customer pays
according to the new plan. Write ordered pairs for the monthly payment when
the customer uses 90, 120, 145, and 150 international minutes in a month.
23. The coordinates
of three vertices of a rectangle are
,
,
and
.
Find the coordinates of the fourth vertex. Then, find the area of the
rectangle.
24. Solve
.
25. Solve
.
26. Solve
.
27. Solve
.
28. Sara needs to
take a taxi to get to the movies. The taxi charges $4.00 for the first mile,
and then $2.75 for each mile after that. If the total charge is $20.50,
then how far was Saras taxi ride to
the movie?
29. Solve
.
30. A video store
charges a monthly membership fee of $7.50, but the charge to rent each movie
is only $1.00 per movie. Another store has no membership fee, but it costs
$2.50 to rent each movie. How many movies need to be rented each month for
the total fees to be the same from either company?
31. Find three
consecutive integers such that twice the greatest integer is 2 less than 3
times the least integer.
32. Solve
for
x.
33. Solve the
proportion
.
34. An architect
built a scale model of a shopping mall. On the model, a circular fountain is
20 inches tall and 22.5 inches in diameter. If the actual fountain is to be
8 feet tall, what is its diameter?
35. Complementary
angles are two angles whose measures add to 90°. The ratio of the measures
of two complementary angles is 4:11. What are the measures of the angles?
36. Find the value
of MN if
cm,
cm,
and
cm.
ABCD
LMNO
37. On a sunny day,
a 5-foot red kangaroo casts a shadow that is 7 feet long. The shadow of a
nearby eucalyptus tree is 35 feet long. Write and solve a proportion to find
the height of the tree.
38. Find 55% of
125.
39. What percent of
74 is 481? If necessary, round your answer to the nearest tenth of a
percent.
40. 66 is 56% of
what number? If necessary, round your answer to the nearest hundredth.
41. Aaron works
part time as a salesperson for an electronics store. He earns $6.75 per hour
plus a percent commission on all of his sales. Last week Aaron worked 17
hours and earned a gross income of $290.63. Find Aarons percent commission
if his total sales for the week were $3,350. If necessary, round your answer
to the nearest hundredth of a percent.
42. Hannah had
dinner at her favorite restaurant. If the sales tax rate is 4% and the sales
tax on the meal came to $1.25, what was the total cost of the meal,
including sales tax and a 20% tip?
43. Find the
percent change from 24 to 72. Tell whether it is a percent increase or
decrease. If necessary, round your answer to the nearest percent.
44. Write the
inequality shown by the graph.
45. Solve the
inequality and graph the solution.
46. Solve the
inequality
£
2 and graph the solutions.
47. Solve and graph
the solutions of the compound inequality
.
48. Solve and graph
the compound inequality.
OR
49. Solve the
inequality
and
graph the solutions. Then write the solutions as a compound inequality.
50. Determine a
relationship between the x- and y-values. Write an equation.
51. Identify the
independent and dependent variables in the situation.
As Kyoko works more hours, her total pay increases.
52. For
,
find
when
x = 1.
53. Graph
for
the domain D: {8, 4, 0, 4, 8}.
54. Determine
whether the sequence appears to be an arithmetic sequence. If so, find the
common difference and the next three terms in the sequence.
5, 11, 17, 23, 29, . . .
55. Find the 20th
term in the arithmetic sequence 4, 1, 6, 11, 16,...
56. Sylvie is going
on vacation. She has already driven 60 miles in one hour. Her average speed
for the rest of the trip is 57 miles per hour. How far will Sylvie have
driven 7 hours later?
57. Find the x-
and y-intercepts of
.
58. Find the slope
of the line that contains
and
.
59. Find the slope
of the line described by x 3y = 6.
60. Graph the line
with the slope
and
y-intercept 2.
61. Write the
equation that describes the line with slope =
and
y-intercept =
in
slope-intercept form.
62. Write the
equation that describes the line in slope-intercept form.
slope = 4, point (3, 2) is on the line
63. Write the
equation
in
slope-intercept form. Then graph the line described by the equation.
64. Write an
equation in point-slope form for the line that has a slope of
and
contains the point (8, 7).
65. The equations
of four lines are given. Identify which lines are parallel.
|
Line 1:
|
y
=
 x
+ 6
|
|
Line 2:
|
x
+
 y
= 6
|
|
Line 3:
|
y
=
 x
8
|
|
Line 4:
|
y
+ 7 =
 (x
+ 4)
|
66. Show that
ABCD is a parallelogram.
67. Show that
LMN is a right triangle.
68. Tell whether
the ordered pair (5, 3) is a solution of the system
.
69. Solve
by
using substitution. Express your answer as an ordered pair.
70. Solve
by
using elimination. Express your answer as an ordered pair.
71. Solve
by
using elimination. Express your answer as an ordered pair.
72. Solve
by
using elimination. Express your answer as an ordered pair.
73. At the local
pet store, zebra fish cost $2.10 each and neon tetras cost $1.85 each. If
Marsha bought 13 fish for a total cost of $25.80, not including tax, how
many of each type of fish did she buy?
74. Simplify
.
75. Simplify
.
76. Evaluate
for
and
.
77. Simplify
.
78. Simplify
.
79. Simplify
.
80. Simplify
.
81. Find the degree
of the polynomial
.
82. Add or
subtract.
83. Subtract.
84. The legs of an
isosceles triangle measure
units.
The perimeter of the triangle is
units.
Write a polynomial that represents the measure of the base of the triangle.
85. Multiply.
86. Multiply.
87. Multiply.
88. Multiply.
89. Factor the
polynomial
.
90. Factor
by
grouping.
91. Factor the
trinomial
.
92. Factor
.
93. Factor
completely.
94. Solve the
quadratic equation
by
factoring.
95. Write a
polynomial to represent the area of the shaded region. Then solve for x
given that the area of the shaded region is 24 square units.
96. Solve
by
using square roots.
97. The monthly
rents for five apartments advertised in a newspaper were $650, $650, $740,
$1650, and $820. Use the mean, median, and mode of the rents to answer the
question. Which value best describes the monthly rents? Explain.
mean = $902, median = $740, mode = $650
98. A manufacturer
inspects 800 personal video players and finds that 798 of them have no
defects. What is the experimental probability that a video player chosen at
random has no defects? Express your answer as a percent.
99. An experiment
consists of rolling a number cube. Find the theoretical probability of
rolling a number greater than 4. Express your answer as a fraction in
simplest form.
100. Find the next
three terms in the geometric sequence
,
6,
,
,
...
101. A computer is
worth $4000 when it is new. After each year it is worth half what it was the
previous year. What will its worth be after 4 years? Round your answer to
the nearest dollar.
102. Simplify the
expression
.
103. Simplify the
expression
.
All variables represent nonnegative numbers.
104. Simplify
.
105. Simplify the
expression
.
106. Multiply.
Write the product in simplest form.
107. Simplify the
quotient
108. Solve the
equation
.
Check your answer.
109. Solve the
equation
.
Check your answer.
110. Solve the
equation
.
Check your answer.
111. Find the
excluded values of the rational expression
.
112. Simplify the
rational expression
.
Identify any excluded values.
113. Simplify the
rational expression
.
114. Multiply.
Simplify your answer.
115. Multiply.
Simplify your answer.
.
116. Divide.
Simplify your answer.
117. Simplify
.
118. Add. Simplify
your answer.
119. Add. Simplify
your answer.
120. Subtract and
simplify. Find the excluded values.
Name ________________________________________________
Answer Section
SHORT ANSWER
1. 14y
y
represents the number of letters Julia wrote.
Think: y groups of 14 letters.
14y
2. 4
Substitute the values for q and v into the expression, and then
subtract.
3. 3
Change the subtraction sign to an addition sign, and change the sign of the
second number.
4. 20
Substitute 4 for u. Then multiply.
5.
|
|
|
|
|
Write
 as
an improper fraction.
|
|
|
To divide by
 multiply
by
 .
|
|
|
Multiply.
|
|
|
Simplify.
|
|
|
|
6. 0
The quotient of 0 and any nonzero number is 0.
7. 81
The exponent tells the number of times to multiply the base number by itself.
The negative sign in front of the expression multiplies the expression by 1.
Multiply 3 by itself 4 times, and then multiply your answer by 1.
8. 16
The exponent tells the number of times to multiply the base number by itself.
Multiply 4 by itself 2 times.
9.
The exponent tells how many times to multiply the fraction by itself.
Multiply
by
itself 2 times.
10.
The number given as a base should be multiplied by itself a certain number of
times in order to represent the value of the whole number given.
The product of two 3s is 9.
11. 1,600 ants
If the population of the ant hill is 40 ants and it doubles every 10 days, then
to find its population in 20 days, make a chart to see what the population is
after a certain number of days.
In 10 days, the population is 40 ants.
In 2 10 days, the population is 402 ants.
In 3 10 days, the population is 403 ants.
In 4 10 days, the population is 404 ants.
12. 14
|
196 =
|
What number squared equals 196?
|
|
=
14
|
The sign to the left of the radical determines whether the square root
is positive or negative.
|
13. 14 ft
202 is between 196 and 225. Since 202 is closer to 196, the best estimate for
the side length is 14 ft.
14. 14
Use the order of operations:
1. Perform operations in parentheses.
2. Evaluate powers.
2. Multiply or divide from left to right.
3. Add or subtract from left to right.
15. 93
Use the order of operations:
1. Perform operations in parentheses.
2. Evaluate powers.
3. Multiply or divide from left to right.
4. Add or subtract from left to right.
16. 14
First, simplify the numerator of the fraction, and then divide the numerator by
the denominator. Next, subtract the terms in the absolute value, and then find
the absolute value.
=
Finally, add the two terms.
=
14
17. $1.90
Use the formula (P + 5N + 10D + 25Q)
Έ
100. Substitute the values from the table.
Tatia has $1.90.
18.
You must use each of the numbers at least once, and you may use any operations.
Pay attention to the order of operations.
19. 11
|
|
|
|
|
Use the Commutative Property.
|
|
|
Use the Associative Property to make groups of compatible numbers.
|
|
|
Simplify.
|
|
|
|
20. 11 40 + 11 7;
517
Rewrite 47 as 40 + 7. Then multiply each term by 11 and add the products.
21.
|
|
|
|
|
Group like terms.
|
|
|
Add or subtract the coefficients.
|
22.
;
(90, 34), (120, 37), (145, 39.5), (150, 40)
Let y represent the monthly payment and x represent the number of
minutes of international calls.
|
monthly payment
|
is
|
$25
|
plus
|
$0.10
|
for each
|
international minute
|
|
y
|
=
|
25
|
+
|
0.10
|
|
x
|
|
Number of international minutes
|
Rule
|
Monthly
payment
|
Ordered pair
|
|
x
(input)
|
|
y
(output)
|
(x, y)
|
|
90
|
|
$34.00
|
(90, 34)
|
|
120
|
|
$37.00
|
(120, 37)
|
|
145
|
|
$39.50
|
(145, 39.5)
|
|
150
|
|
$40.00
|
(150, 40)
|
23.
;
Area = 72 square units
Step 1
Plot the points.
Step 2
Find the fourth vertex.
The fourth vertex will have the same x-coordinate as C(10,3) and
the same y-coordinate as A(1, 5).
x-coordinate:
10
y-coordinate:
5
The fourth vertex is D(10, 5).
Step 3
Find the area of the rectangle.
square
units
24. q = 205
|
|
|
|
|
Since q is divided by 5, multiply both sides by 5 to undo the
division.
|
|
q
= 205
|
|
Check:
|
|
|
|
|
To check your solution, substitute 205 for q in the original
equation.
|
|
|
|
25. a = 15
|
|
First x is multiplied by 2. Then 14 is added.
|
|
|
Work backward: Subtract 14 from both sides.
|
|
|
|
|
|
Since x is multiplied by 2, divide both sides by 2 to undo the
multiplication.
|
|
|
|
26.
|
|
|
|
|
Since
 is
subtracted from
 ,
add
to
both sides to undo the subtraction.
|
|
|
|
|
|
Since f is divided by 45, multiply both sides by 45 to undo the
division.
|
|
|
Simplify.
|
27.
|
|
|
|
|
Use the Commutative Property of Addition.
|
|
|
Combine like terms.
|
|
|
Since 10 is added to 17a, subtract 10 from both sides to undo the
addition.
|
|
|
|
|
|
Since a is multiplied by 17, divide both sides by 17 to undo the
multiplication.
|
|
|
|
28. 7 miles
Let d be the distance (in miles) to the movies, then
is
the number of miles after the first mile. So a formula for the total charge
could be
|
first mile charge
|
+
|
|
|
rate after first mile
|
=
|
total charge
|
|
|
4.00
|
+
|
|
|
2.75
|
=
|
20.50
|
Subtract 4.00 from each side.
|
|
|
|
|
|
2.75
|
=
|
20.50
4.00
|
|
|
|
|
|
|
2.75
|
=
|
16.5
|
Divide both sides by 2.75.
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
=
|
6
|
Add 1 to both sides.
|
|
|
|
|
|
d
|
=
|
6 + 1
|
|
|
|
|
|
|
d
|
=
|
7
|
|
29. n =
|
|
|
|
|
Combine like terms.
|
|
|
Add to undo the subtraction. Or subtract to undo the addition. Then,
divide to undo the multiplication.
|
|
n
=
|
|
30. 5 movies
Let m represent the number of movies rented each month.
Here are the costs for each company (in dollars).
To collect the variable terms on one side, subtract m from both sides.
|
7.5 m
|
=
|
2.5m m
|
|
7.5
|
=
|
1.5 m
|
Divide both sides by 1.5.
|
|
=
|
m
|
|
5
|
=
|
m
|
31. 6, 7, 8
Let g represent the greatest integer.
The expressions for the three consecutive integers from least to greatest:
,
,
g.
|
twice
|
the greatest integer
|
|
3 times
|
the least integer
|
|
|
g
|
|
|
(
)
|
|
|
To create an equation, use the additional data that 2g is 2 less
than
 .
|
|
|
Solve the equation.
|
The three consecutive numbers are 6, 7, and 8.
32.
|
|
Add z to both sides.
Divide both sides by 4.
|
33. x = 25
|
|
|
|
|
Use cross products.
|
|
|
Divide both sides by 6.
|
|
|
|
34. 9 ft
|
|
Write the scale as a fraction.
|
|
|
Let x be the actual diameter.
|
|
|
Use cross products to solve.
|
|
|
|
35. 24°, 66°
Let a represent the measure of one of the complementary angles and
represent
the measure of the second angle.
|
ratio of the measures of the angles
|
is
|
4:11
|
|
|
=
|
|
Solve
.
|
|
Use cross products.
|
|
|
Distribute.
|
|
|
Add 4a to both sides.
|
|
|
Simplify.
|
|
|
Solve the equation.
|
|
|
|
|
|
Substitute 24 for a to find the measure of the second angle.
|
The measures of the complementary angles are 24° and 66°.
36. 22.4 cm
A
corresponds to L, B corresponds to M, C corresponds
to N, and D corresponds to O.
|
|
|
|
|
Use cross products.
|
|
|
Since x is multiplied by 21, divide both sides by 21 to undo the
multiplication.
|
|
|
|
|
|
|
|
_MN is 22.4 cm.
|
|
37.
;
25 feet
|
|
|
|
|
Use cross products.
|
|
|
Since x is multiplied by 7, divide both sides by 7 to undo the
multiplication.
|
|
|
|
|
|
|
|
The tree is 25 feet tall.
|
|
38. 68.75
Method 1
Use a proportion.
|
|
Use the percent proportion.
|
|
|
Let x represent the part.
|
|
|
Find the cross products.
|
|
|
Since x is multiplied by 100, divide both sides by 100 to undo
the multiplication.
|
|
|
|
|
_55% of 125 is 68.75.
|
|
Method 2
Use an equation.
|
|
Write an equation. Let x represent the part.
|
|
|
Write the percent as a decimal and multiply.
|
|
|
|
|
55% of 125 is 68.75.
|
|
39. 650%
Method 1
Use a proportion.
|
|
Use the percent proportion.
|
|
|
Let x represent the percent.
|
|
|
Find the cross products.
|
|
|
Since x is multiplied by 74, divide both sides by 74 to undo the
multiplication.
|
|
|
|
|
|
|
|
_481 is 650% of 74.
|
|
Method 2
Use an equation.
|
|
Write an equation. Let x represent the percent.
|
|
|
Since x is multiplied by 74, divide both sides by 74 to undo the
multiplication.
|
|
|
The answer is a decimal.
|
|
|
Write the decimal as a percent.
|
|
|
|
|
_481 is 650% of 74.
|
|
40. 117.86
Method 1
Use a proportion.
|
|
Use the percent proportion.
|
|
|
Let x represent the whole.
|
|
|
Find the cross products.
|
|
|
Since x is multiplied by 56, divide both sides by 56 to undo the
multiplication.
|
|
|
|
|
|
|
|
_56% of 66 is 117.86.
|
|
Method 2
Use an equation.
|
|
Write an equation. Let x represent the whole.
|
|
|
Write the percent as a decimal.
|
|
|
Since x is multiplied by 0.56, divide both sides by 0.56 to undo
the multiplication.
|
|
|
|
|
|
|
|
_56% of 66 is 117.86.
|
|
41. 5.25%
Write the formula for gross income.
gross income
(income
number
of hours)
commission
Write the formula for commission.
gross income
(income
number
of hours)
%
of total sales
|
|
Substitute value given in the problem. Let x represent the
percent commission.
|
|
|
Multiply
|
|
|
Subtract.
|
|
|
Since x is multiplied by 3,350, divide both sides by 3,350 to
undo the multiplication.
|
|
|
The answer is a decimal.
|
|
_5.25% = x
Write
the decimal as a percent.
|
|
|
|
|
_Aarons percent commission is 5.25%.
|
42. $38.75
Step 1
Find the cost of the meal before the tip and sales tax.
|
|
Write the formula for the sales tax.
|
|
|
Substitute known values.
|
|
|
Solve for c, the cost of the meal.
|
Step 2
Find the total cost of the meal,
including tip and sales tax.
|
|
Write the formula for the total cost.
|
|
|
|
|
|
Substitute the known values.
|
|
|
|
|
|
|
|
|
|
43. 200% increase
|
|
|
|
|
Substitute the given values.
|
|
|
|
If the first number is less than the second number, there is a percent of
increase.
If the first number is greater than the second number, there is a percent of
decrease.
44. m
³
3
Use the variable m. The arrow points to the right, so use either > or
³.
The solid circle at 3 means that 3 is a solution, so use
³.
45.
Step 1:
Rewrite both mixed numbers as improper fractions.
and
Step 2:
Solve the inequality.
|
|
Rewrite the inequality.
|
|
|
Subtract
 from
both sides.
|
|
|
Rewrite the fractions with a common denominator.
|
|
 =
|
Simplify.
|
Step 3:
Graph the inequality.
46. z
³
8
|
£
2
|
|
|
 ³
2(4)
|
Multiply both sides by 4 to isolate z. When you multiply by a
negative number, reverse the inequality symbol.
|
|
z
³
8
|
|
Use a solid circle when the value is included in the graph, such as with
³
or
£.
Use an empty circle when the value is not included, such as with > or <.
47.
AND
|
|
AND
|
|
Write the compound inequality using AND.
|
|
|
|
|
Solve each simple inequality.
|
|
|
|
|
Divide to undo the multiplication.
|
|
|
AND
|
|
|
First, graph the solutions of each simple inequality. Then, graph the
intersection by finding where the two graphs overlap.
48.
OR
First solve each simple inequality to obtain
OR
.
The graph of the compound inequality is the union of the graph of
and
the graph of
.
Find the union by combining the two regions.
49.
|
|
|
|
|
Add 9 to isolate the absolute-value expression.
|
|
|
Think: What numbers have an absolute value less than 8?
|
|
 AND
|
 is
between 8 and 8, inclusive.
|
|
 AND
|
Solve the two inequalities.
|
|
|
Write the solution as a compound inequality.
|
50. y = x
+ 3
The correct equation is y = x + 3.
51. Independent: hours
worked; Dependent: total pay
The value of the dependent variable depends on the value of the
independent variable.
In this situation, the total amount Kyoko is paid depends on the number
of hours she works, so hours worked is the independent variable and total pay is
the dependent variable.
52. 6
|
|
|
|
|
Substitute 1 for x.
|
|
|
Simplify.
|
53.
Step 1:
Solve the equation for y: y =
x
+
.
Step 2:
Substitute the x-values of the given domain into the equation to find the
y-values. Make a table to help organize your work.
|
x
|
y
=
x
+
|
(x, y)
|
Step 3:
Graph the ordered pairs.
|
|
8
|
y
=
(8)
+
=
3
|
(8, 3)
|
|
4
|
y
=
(4)
+
=
1
|
(4, 1)
|
|
0
|
y
=
(0)
+
=
1
|
(0, 1)
|
|
4
|
y
=
(4)
+
=
3
|
(4, 3)
|
|
8
|
y
=
(8)
+
=
5
|
(8, 5)
|
54. Yes; common
difference 6; next 3 terms are 35, 41, 47
For a sequence to be an arithmetic sequence, each number subtracted from the one
before it should result in a common difference.
This sequence is arithmetic. Each term differs from the previous one by 6.
55. 91
Find a specific term from a given sequence by using the equation
,
where:
an
= your result
a1
= the initial term of the sequence
n
= the number in the sequence you want to calculate
d
= the common difference between the terms
n
is given in the problem, a1 is the first term in the sequence,
and d is the difference between adjacent terms.
56. 459 miles
Since you want to find the distance for 7 hours later, you need to find the 8th
term of the sequence. So, n = 8.
|
|
Write the rule to find the nth term.
|
|
an
= 60 + (8 1)(57)
|
Substitute 60 for
 ,
57 for d, and 8 for n.
|
|
an
= 459
|
Simplify.
|
Sylvie will have traveled 459 miles.
57. x-intercept:
8, y-intercept: 4
To find the x-intercept, replace y with 0 and solve for x;
to find the y-intercept, replace x with 0 and solve for y.
|
x-intercept
|
y-intercept
|
|
|
|
58.
|
|
Use the slope formula.
|
|
|
Substitute
for
 and
for
 .
|
|
 =
|
Simplify.
|
59.
Find the x-intercept by substituting x = 0 into the equation. Find
the y-intercept by substituting y = 0 into the equation. Use the
two intercept points and the slope formula,
,
to calculate the slope.
60.
Plot the y-intercept 2 on the graph at (0, 2). The slope is
,
so from the y-intercept, rise 1 units and run 3 units. Plot another
point. Connect the points to graph the line.
61. y =
x
+
The slope-intercept form is y = mx + b, where m is
the slope and b is the y-intercept. Substituting
for
the slope and
for
the y-intercept gives y =
x
+
.
62. y = 4x
14
If you are given the slope and one point, you can find the y-intercept by
substituting for m, x, and y in the equation y = mx
+ b. Then, solve for b.
2 = 4(3) + b
2 = 12 + b
14 = b
So, the equation of the line in slope-intercept form is y = 4x
14.
63.
,
Plot
.
Count 1 down and 2 right, and plot another point. Draw a line connecting the two
points.
64. y + 7 =
(x
+ 8)
Substitute the point and slope into the point-slope form
,
where m represents the slope and
represents
a point on the line.
65. Lines 2 and 3 are
parallel.
Write all the equations in slope-intercept form (y = mx + b). The
equations that have the same slope but different y-intercepts are
parallel lines.
66.
ABCD
is a parallelogram because both pairs of opposite sides are parallel.
Use the ordered pairs and the slope formula to find the slopes of the four line
segments formed by the four points given. If the points given create two sets of
parallel segments, then the quadrilateral formed by the four points is a
parallelogram.
|
|
The formula for the slope between two points
(x1, y1) and (x2,
y2) is
|
67. Slope of
and
slope of
.
is
perpendicular to
because
.
LMN is a right triangle because it contains a right angle.
A right triangle contains one right angle.
L
and
N
are not right angles, so the only possibility is
M.
If LMN is a right triangle,
will
be perpendicular to
.
slope of
slope of
,
so
is
perpendicular to
.
LMN
is a right triangle because it contains a right angle.
68. yes
Substitute 5 for x and 3 for y in both equations. Since these
values make both equations true, (5, 3) is a solution of the system.
69. (
,
)
|
Step 1
|
|
The second equation is solved for y.
|
|
|
|
|
|
Step 2
|
|
|
|
|
|
Substitute
 for
y in the first equation.
|
|
|
|
|
|
Step 3
|
|
Simplify and solve for x.
|
|
|
|
|
|
|
|
Divide both sides by 4.
|
|
|
x
=
|
|
|
|
|
|
|
Step 4
|
|
Write one of the original equations.
|
|
|
y =
|
Substitute
for
x.
|
|
|
|
Find the value of y.
|
|
|
|
|
|
|
(
,
)
|
Write the solution as an ordered pair.
|
70. (0, 2)
|
Step 1
|
3x 6y = 12
|
|
|
|
2x + 6y = 12
|
The y-terms have opposite coefficients.
|
|
|
5x = 0
|
Add the equations to eliminate the y terms.
|
|
|
x
= 0
|
|
|
|
|
|
|
Step 2
|
3(0) 6y = 12
|
Substitute for x in one of the original equations.
|
|
|
0 6y = 12
|
Simplify and solve for y.
|
|
|
6y = 12
|
|
|
|
y
= 2
|
|
|
|
|
|
|
|
(0, 2)
|
Write the solution as an ordered pair.
|
71. (5, 0)
|
|
Multiply all expressions in the second equation by
.
|
|
|
Add the two equations together.
|
|
|
Divide both sides by 2.
|
|
|
Solve for x.
|
|
|
|
|
|
Substitute the value for x into one of the original equations and
solve for y.
|
|
|
|
72. (4, 3)
First, multiply each equation by a number to get opposite coefficients.
|
|
|
Multiply the first equation by 3
|
|
|
|
and the second by 5 to get opposite
|
|
|
|
y-coefficients.
|
|
Step 1
|
|
Add the two equations to eliminate the y-term.
|
|
|
|
|
|
Step 2
|
|
Simplify and solve for x.
|
|
|
x
= 4
|
|
|
|
|
|
|
Step 3
|
|
Write one of the original equations.
|
|
|
|
Substitute 4 for x. Simplify and solve for y.
|
|
|
|
|
|
|
|
|
|
|
y
= 3
|
|
73. 7 zebra fish, 6
neon tetras
Let z be the number of zebra fish and let n be the number of neon
tetras that Marsha bought. Then solve the following system of equations.
|
|
Marsha spent $25.80.
Marsha bought 13 fish.
|
|
|
Multiply the second equation by 2.10
|
|
|
Add the two equations to eliminate the z term.
Solve for n.
|
To solve for z, substitute 6 for n in the first equation.
|
|
Simplify.
Solve for z.
|
74.
|
=
|
The reciprocal of 2 is
 .
|
|
=
|
 =
8.
|
75. 1
Any nonzero base to the zero power is equal to 1.
=
1
76.
|
|
Substitute 3 for a and 3 for b.
|
|
(
)(1)
|
Evaluate expressions with exponents.
|
|
|
Simplify.
|
77.
To multiply powers with the same base, keep the same base and add the exponents.
=
=
78.
|
|
Use the Power of a Power Property.
|
|
|
Simplify the exponent of the first term.
|
|
|
Add the exponents since the powers have the same base.
|
|
|
|
|
|
Write with a positive exponent.
|
79.
|
|
=
|
Simplify exponents with like bases:
 .
|
|
|
=
|
Use the Power of a Quotient Property.
|
|
|
=
|
Use the Power of a Product Property.
|
|
|
=
|
Simplify:
 .
|
|
|
=
|
Use the Power of a Power Property to simplify the exponents.
|
80.
|
=
|
Rewrite with a positive exponent.
|
|
 =
 =
|
Use the Power of a Quotient Property, and simplify.
|
81. 9
:
degree 9,
:
degree 2, and
:
degree 3
The degree of the polynomial is 9.
82.
|
|
= 10m + 2m4 13m
20m4
|
Identify like terms.
|
|
|
= 10m 13m + 2m4
20m4
|
Use the Commutative Property to move like terms together.
|
|
|
=
|
Combine like terms.
|
83.
|
|
|
|
=
|
Rewrite subtraction as addition of the opposite.
|
|
=
|
Identify like terms. Rearrange terms to get like terms together.
|
|
=
|
Combine like terms.
|
84.
The perimeter of a triangle is the sum of the measures of its sides. In an
isosceles triangle, the legs have equal lengths.
|
|
|
|
 +
base
|
Substitute the given values.
|
|
|
Combine like terms.
|
|
|
Subtract
 from
both sides.
|
|
|
Subtract
 from
both sides.
|
|
|
Add 2 to both sides.
|
85.
|
|
Rearrange the terms to group like bases.
|
|
|
To multiply powers, add the exponents.
|
86.
|
|
|
|
|
Distribute
 and
 .
|
|
|
Distribute
and
again.
|
|
|
Multiply.
|
|
|
Combine like terms.
|
87.
|
Method 1
 =
 +
|
The factors fit the pattern for squaring a binomial to get a perfect
square trinomial. Use the rule for
.
|
|
|
Identify
 and
 from
the given binomial.
|
|
|
Use these values to determine
,
 ,
and
.
|
|
|
Substitute the terms into the corresponding places.
|
|
|
|
|
Method 2
|
Use FOIL to multiply the binomials.
|
88.
|
|
|
|
|
Use the rule for
 .
|
|
|
Use the FOIL method, and then combine like terms.
|
|
|
Simplify.
|
89.
|
|
Find the GCF. The GCF of
 ,
 ,
and
 is
|
|
|
Write the terms as products using the GCF.
|
|
|
Use the Distributive Property to factor out the GCF.
|
90.
|
|
|
|
|
Group terms that have a common number or variable as a factor.
|
|
|
Factor out the GCF of each group.
|
|
|
Factor out the common factor
 .
|
91.
Find two factors of 625 such that their sum is 50. These factors are 25 and 25.
.
Simplify the expression to
.
92.
Try factors of 3 for the coefficients and factors of 8 for the constant terms.
The combination that works is:
=
=
93.
Factor out the GCF, 3z. The remaining polynomial,
,
is a perfect square trinomial that can be factored.
94. 4 and 2
|
|
|
|
|
Factor the trinomial.
|
|
 or
|
Use the Zero Product Property.
|
|
 or
|
Solve each equation.
|
The solutions are 4 and 2.
95.
Step 1
The area of the shaded region is equal to the area of the large rectangle minus
the area of the small rectangle.
|
|
Write an equation.
|
|
|
Use FOIL.
|
|
|
Simplify.
|
Step 2
|
|
Rewrite the equation in standard form.
|
|
|
Factor.
|
|
 or
|
Use the Zero Product Property.
|
|
 or
|
Solve each equation.
|
|
|
A negative number is not reasonable for distance.
|
96. There is no
solution.
Take the square root of both sides of the equation. There is no number whose
square is negative, so there is no solution.
97. The median best
describes the rents because most of the rents were near $740.
The median, or middle value, of a data set is usually the best description of
the values.
98. 99.75%
Divide the number of video players with no defects by the total number of
players. Multiply the result by 100 to express the answer as a percent.
99.
There are six possible outcomes when a fair number cube is rolled. Because the
number cube is fair, all outcomes are equally likely. There are two numbers
greater than 4 on the number cube: 5 and 6. So the probability of rolling one of
these numbers is
.
100.
,
,
Find the value of the common ratio by dividing each term by the one before it.
Then, multiply each term by the common ratio to get the next term.
101. $250
Use the equation an = a1 r n 1
to solve the problem.
The variables represent the following values:
an
= value of the computer after n years
a1
= the initial value of the computer
r
= rate of depreciation, 0.5
n
= the number of years
102.
103.
Factor perfect squares out of the radicand. Use the Product Property of Square
Roots to take the square root of each factor separately. Simplify.
104.
|
|
|
|
=
|
Use the Quotient Property.
|
|
=
|
Find perfect square factors if possible. Write 300 as 100(3).
|
|
=
|
Use the Product Property.
|
|
=
|
Simplify.
|
105.
|
|
|
|
=
|
Factor the radicands using perfect-square factors.
|
|
=
|
Use the Product Property of Square Roots.
|
|
=
|
|
|
=
|
Combine like radicals.
|
106.
+
Distribute
.
Use the Product Property of Square Roots to multiply the factors in each term.
If the radicand in either term contains any perfect square factors, factor the
radicand(s) and simplify. Combine like terms if applicable.
107.
|
|
|
Multiply by a form of 1 to get a perfect-square radicand in the
denominator.
|
|
|
|
Simplify the denominator.
|
|
|
|
|
108. z = 676
|
|
|
|
|
Add 10 to both sides.
|
|
|
Square both sides.
|
|
|
|
109.
81
There are two methods to solve the equation.
Method 1
Divide both sides by 8 and then square both sides.
Method 2
Square both sides and then divide both sides by (8)2 which is 64.
110. z = 3
|
|
|
|
|
Add
 to
both sides.
|
|
|
Square both sides.
|
|
|
Simplify.
|
|
z
= 3
|
Solve.
|
111. The excluded values are 4
and 1.
The excluded values are all values of n that make the denominator equal
to zero.
|
n2
5n + 4 = 0
|
Set the denominator equal to 0.
|
|
(n 4)(n 1) = 0
|
Factor.
|
|
n
4 =0 or n 1 = 0
|
Use the Zero-Product Property to solve each factor for n.
|
|
n
= 4 or n = 1
|
4 must be excluded. 1 must be excluded.
|
The excluded values are 4 and 1.
112. 3r; r
3
Factor common factors out of the numerator and/or denominator. Divide out the
common factors to simplify the expression. Finally, use the original denominator
to determine excluded factors.
113.
|
|
|
|
=
|
Factor.
|
|
=
|
Identify the opposite binomials.
|
|
=
|
Divide out the common factors, and simplify.
|
114.
Arrange the expressions so like terms are together:
.
Multiply the numerators and denominators, remembering to add exponents when
multiplying:
.
Divide, remembering to subtract exponents:
.
Since
,
this expression simplifies to
115.
|
|
|
|
=
|
Factor the numerator and denominator.
|
|
=
|
Simplify.
|
|
=
|
Multiply the remaining factors.
|
116.
|
|
|
|
=
|
Write as multiplication by the reciprocal.
|
|
=
|
Multiply the numerators and the denominators.
|
|
=
|
Divide out common factors. Simplify.
|
117.
|
|
|
|
=
|
Factor.
|
|
=
|
Multiply the numerators and the denominators.
|
|
=
|
Divide out common factors and simplify.
|
118.
|
|
|
|
|
Combine like terms in the numerator.
|
|
|
Factor. Divide out common factors.
|
|
|
Simplify.
|
119.
|
|
Identify the LCD,
 .
|
|
=
|
|
|
=
|
Rewrite each fraction with a denominator of
.
|
|
=
|
Add.
|
|
=
|
 and
 are
not like terms, so they cannot be combined. Factor and divide out common
factors.
|
|
=
|
Simplify.
|
120.
;
and
|
|
|
|
=
|
Factor the difference of squares.
|
|
=
|
Write each expression using the LCD.
|
|
=
|
Subtract the numerators.
|
|
=
|
Distribute.
|
|
=
|
Simplify.
|
Excluded values are values that make the denominator equal to 0. If
or
,
then the denominator is 0. Therefore,
and
.
Name ________________________________________________
Short Answer: Show all work and
leave answers in fraction form.
1. Julia
wrote 14 letters to friends each month for y months in a row. Write
an expression to show how many total letters Julia wrote.
2. Evaluate the
expression q v for q = 5 and v = 1.
3. Subtract.
5 (8)
4. Evaluate 5u
for u = 4.
5. Divide.
6. Divide.
0
Έ
5.928
7. Simplify
.
8. Simplify
.
9. Simplify
.
10. Write 9 as a
power of the base 3.
11. If the
population of an ant hill doubles every 10 days and there are currently 40
ants living in the ant hill, what will the ant hill population be in 20
days?
12. Find the square
root.
13. The area of a
square garden is 202 square feet. Estimate the side length of the garden.
14. Simplify
.
15. Simplify
.
16. Simplify the
expression
.
17. Tatia has coins
in pennies, nickels, dimes, and quarters. The total amount of money she has
in dollars can be found using the expression (P + 5N + 10D
+ 25Q)
Έ
100. Use the table to find how much money Tatia has.
18. Use the numbers
2, 3, 5, and 8 to write an expression that has a value of
.
You may use any operations, and you must use each of the numbers at least
once.
19. Simplify the
expression
.
20. Write 11 47
using the Distributive Property. Then simplify.
21. Simplify by
combining like terms.
22. A phone company
advertises a new plan in which the customer pays a fixed amount of $25 per
month for unlimited calls in the country, and $0.10 per minute for
international calls. Find a rule for the monthly payment a customer pays
according to the new plan. Write ordered pairs for the monthly payment when
the customer uses 90, 120, 145, and 150 international minutes in a month.
23. The coordinates
of three vertices of a rectangle are
,
,
and
.
Find the coordinates of the fourth vertex. Then, find the area of the
rectangle.
24. Solve
.
25. Solve
.
26. Solve
.
27. Solve
.
28. Sara needs to
take a taxi to get to the movies. The taxi charges $4.00 for the first mile,
and then $2.75 for each mile after that. If the total charge is $20.50,
then how far was Saras taxi ride to
the movie?
29. Solve
.
30. A video store
charges a monthly membership fee of $7.50, but the charge to rent each movie
is only $1.00 per movie. Another store has no membership fee, but it costs
$2.50 to rent each movie. How many movies need to be rented each month for
the total fees to be the same from either company?
31. Find three
consecutive integers such that twice the greatest integer is 2 less than 3
times the least integer.
32. Solve
for
x.
33. Solve the
proportion
.
34. An architect
built a scale model of a shopping mall. On the model, a circular fountain is
20 inches tall and 22.5 inches in diameter. If the actual fountain is to be
8 feet tall, what is its diameter?
35. Complementary
angles are two angles whose measures add to 90°. The ratio of the measures
of two complementary angles is 4:11. What are the measures of the angles?
36. Find the value
of MN if
cm,
cm,
and
cm.
ABCD
LMNO
37. On a sunny day,
a 5-foot red kangaroo casts a shadow that is 7 feet long. The shadow of a
nearby eucalyptus tree is 35 feet long. Write and solve a proportion to find
the height of the tree.
38. Find 55% of
125.
39. What percent of
74 is 481? If necessary, round your answer to the nearest tenth of a
percent.
40. 66 is 56% of
what number? If necessary, round your answer to the nearest hundredth.
41. Aaron works
part time as a salesperson for an electronics store. He earns $6.75 per hour
plus a percent commission on all of his sales. Last week Aaron worked 17
hours and earned a gross income of $290.63. Find Aarons percent commission
if his total sales for the week were $3,350. If necessary, round your answer
to the nearest hundredth of a percent.
42. Hannah had
dinner at her favorite restaurant. If the sales tax rate is 4% and the sales
tax on the meal came to $1.25, what was the total cost of the meal,
including sales tax and a 20% tip?
43. Find the
percent change from 24 to 72. Tell whether it is a percent increase or
decrease. If necessary, round your answer to the nearest percent.
44. Write the
inequality shown by the graph.
45. Solve the
inequality and graph the solution.
46. Solve the
inequality
£
2 and graph the solutions.
47. Solve and graph
the solutions of the compound inequality
.
48. Solve and graph
the compound inequality.
OR
49. Solve the
inequality
and
graph the solutions. Then write the solutions as a compound inequality.
50. Determine a
relationship between the x- and y-values. Write an equation.
51. Identify the
independent and dependent variables in the situation.
As Kyoko works more hours, her total pay increases.
52. For
,
find
when
x = 1.
53. Graph
for
the domain D: {8, 4, 0, 4, 8}.
54. Determine
whether the sequence appears to be an arithmetic sequence. If so, find the
common difference and the next three terms in the sequence.
5, 11, 17, 23, 29, . . .
55. Find the 20th
term in the arithmetic sequence 4, 1, 6, 11, 16,...
56. Sylvie is going
on vacation. She has already driven 60 miles in one hour. Her average speed
for the rest of the trip is 57 miles per hour. How far will Sylvie have
driven 7 hours later?
57. Find the x-
and y-intercepts of
.
58. Find the slope
of the line that contains
and
.
59. Find the slope
of the line described by x 3y = 6.
60. Graph the line
with the slope
and
y-intercept 2.
61. Write the
equation that describes the line with slope =
and
y-intercept =
in
slope-intercept form.
62. Write the
equation that describes the line in slope-intercept form.
slope = 4, point (3, 2) is on the line
63. Write the
equation
in
slope-intercept form. Then graph the line described by the equation.
64. Write an
equation in point-slope form for the line that has a slope of
and
contains the point (8, 7).
65. The equations
of four lines are given. Identify which lines are parallel.
|
Line 1:
|
y
=
x
+ 6
|
|
Line 2:
|
x
+
y
= 6
|
|
Line 3:
|
y
=
x
8
|
|
Line 4:
|
y
+ 7 =
(x
+ 4)
|
66. Show that
ABCD is a parallelogram.
67. Show that
LMN is a right triangle.
68. Tell whether
the ordered pair (5, 3) is a solution of the system
.
69. Solve
by
using substitution. Express your answer as an ordered pair.
70. Solve
by
using elimination. Express your answer as an ordered pair.
71. Solve
by
using elimination. Express your answer as an ordered pair.
72. Solve
by
using elimination. Express your answer as an ordered pair.
73. At the local
pet store, zebra fish cost $2.10 each and neon tetras cost $1.85 each. If
Marsha bought 13 fish for a total cost of $25.80, not including tax, how
many of each type of fish did she buy?
74. Simplify
.
75. Simplify
.
76. Evaluate
for
and
.
77. Simplify
.
78. Simplify
.
79. Simplify
.
80. Simplify
.
81. Find the degree
of the polynomial
.
82. Add or
subtract.
83. Subtract.
84. The legs of an
isosceles triangle measure
units.
The perimeter of the triangle is
units.
Write a polynomial that represents the measure of the base of the triangle.
85. Multiply.
86. Multiply.
87. Multiply.
88. Multiply.
89. Factor the
polynomial
.
90. Factor
by
grouping.
91. Factor the
trinomial
.
92. Factor
.
93. Factor
completely.
94. Solve the
quadratic equation
by
factoring.
95. Write a
polynomial to represent the area of the shaded region. Then solve for x
given that the area of the shaded region is 24 square units.
96. Solve
by
using square roots.
97. The monthly
rents for five apartments advertised in a newspaper were $650, $650, $740,
$1650, and $820. Use the mean, median, and mode of the rents to answer the
question. Which value best describes the monthly rents? Explain.
mean = $902, median = $740, mode = $650
98. A manufacturer
inspects 800 personal video players and finds that 798 of them have no
defects. What is the experimental probability that a video player chosen at
random has no defects? Express your answer as a percent.
99. An experiment
consists of rolling a number cube. Find the theoretical probability of
rolling a number greater than 4. Express your answer as a fraction in
simplest form.
100. Find the next
three terms in the geometric sequence
,
6,
,
,
...
101. A computer is
worth $4000 when it is new. After each year it is worth half what it was the
previous year. What will its worth be after 4 years? Round your answer to
the nearest dollar.
102. Simplify the
expression
.
103. Simplify the
expression
.
All variables represent nonnegative numbers.
104. Simplify
.
105. Simplify the
expression
.
106. Multiply.
Write the product in simplest form.
107. Simplify the
quotient
108. Solve the
equation
.
Check your answer.
109. Solve the
equation
.
Check your answer.
110. Solve the
equation
.
Check your answer.
111. Find the
excluded values of the rational expression
.
112. Simplify the
rational expression
.
Identify any excluded values.
113. Simplify the
rational expression
.
114. Multiply.
Simplify your answer.
115. Multiply.
Simplify your answer.
.
116. Divide.
Simplify your answer.
117. Simplify
.
118. Add. Simplify
your answer.
119. Add. Simplify
your answer.
120. Subtract and
simplify. Find the excluded values.
Name ________________________________________________
Answer Section
SHORT ANSWER
1. 14y
y
represents the number of letters Julia wrote.
Think: y groups of 14 letters.
14y
2. 4
Substitute the values for q and v into the expression, and then
subtract.
3. 3
Change the subtraction sign to an addition sign, and change the sign of the
second number.
4. 20
Substitute 4 for u. Then multiply.
5.
|
|
|
|
|
Write
as
an improper fraction.
|
|
|
To divide by
multiply
by
.
|
|
|
Multiply.
|
|
|
Simplify.
|
6. 0
The quotient of 0 and any nonzero number is 0.
7. 81
The exponent tells the number of times to multiply the base number by itself.
The negative sign in front of the expression multiplies the expression by 1.
Multiply 3 by itself 4 times, and then multiply your answer by 1.
8. 16
The exponent tells the number of times to multiply the base number by itself.
Multiply 4 by itself 2 times.
9.
The exponent tells how many times to multiply the fraction by itself.
Multiply
by
itself 2 times.
10.
The number given as a base should be multiplied by itself a certain number of
times in order to represent the value of the whole number given.
The product of two 3s is 9.
11. 1,600 ants
If the population of the ant hill is 40 ants and it doubles every 10 days, then
to find its population in 20 days, make a chart to see what the population is
after a certain number of days.
In 10 days, the population is 40 ants.
In 2 10 days, the population is 402 ants.
In 3 10 days, the population is 403 ants.
In 4 10 days, the population is 404 ants.
12. 14
|
196 =
|
What number squared equals 196?
|
|
=
14
|
The sign to the left of the radical determines whether the square root
is positive or negative.
|
13. 14 ft
202 is between 196 and 225. Since 202 is closer to 196, the best estimate for
the side length is 14 ft.
14. 14
Use the order of operations:
1. Perform operations in parentheses.
2. Evaluate powers.
2. Multiply or divide from left to right.
3. Add or subtract from left to right.
15. 93
Use the order of operations:
1. Perform operations in parentheses.
2. Evaluate powers.
3. Multiply or divide from left to right.
4. Add or subtract from left to right.
16. 14
First, simplify the numerator of the fraction, and then divide the numerator by
the denominator. Next, subtract the terms in the absolute value, and then find
the absolute value.
=
Finally, add the two terms.
=
14
17. $1.90
Use the formula (P + 5N + 10D + 25Q)
Έ
100. Substitute the values from the table.
Tatia has $1.90.
18.
You must use each of the numbers at least once, and you may use any operations.
Pay attention to the order of operations.
19. 11
|
|
|
|
|
Use the Commutative Property.
|
|
|
Use the Associative Property to make groups of compatible numbers.
|
|
|
Simplify.
|
|
|
|
20. 11 40 + 11 7;
517
Rewrite 47 as 40 + 7. Then multiply each term by 11 and add the products.
21.
|
|
|
|
|
Group like terms.
|
|
|
Add or subtract the coefficients.
|
22.
;
(90, 34), (120, 37), (145, 39.5), (150, 40)
Let y represent the monthly payment and x represent the number of
minutes of international calls.
|
monthly payment
|
is
|
$25
|
plus
|
$0.10
|
for each
|
international minute
|
|
y
|
=
|
25
|
+
|
0.10
|
|
x
|
|
Number of international minutes
|
Rule
|
Monthly
payment
|
Ordered pair
|
|
x
(input)
|
|
y
(output)
|
(x, y)
|
|
90
|
|
$34.00
|
(90, 34)
|
|
120
|
|
$37.00
|
(120, 37)
|
|
145
|
|
$39.50
|
(145, 39.5)
|
|
150
|
|
$40.00
|
(150, 40)
|
23.
;
Area = 72 square units
Step 1
Plot the points.
Step 2
Find the fourth vertex.
The fourth vertex will have the same x-coordinate as C(10,3) and
the same y-coordinate as A(1, 5).
x-coordinate:
10
y-coordinate:
5
The fourth vertex is D(10, 5).
Step 3
Find the area of the rectangle.
square
units
24. q = 205
|
|
|
|
|
Since q is divided by 5, multiply both sides by 5 to undo the
division.
|
|
q
= 205
|
|
Check:
|
|
|
|
|
To check your solution, substitute 205 for q in the original
equation.
|
|
|
|
25. a = 15
|
|
First x is multiplied by 2. Then 14 is added.
|
|
|
Work backward: Subtract 14 from both sides.
|
|
|
|
|
|
Since x is multiplied by 2, divide both sides by 2 to undo the
multiplication.
|
|
|
|
26.
|
|
|
|
|
Since
is
subtracted from
,
add
to
both sides to undo the subtraction.
|
|
|
|
|
|
Since f is divided by 45, multiply both sides by 45 to undo the
division.
|
|
|
Simplify.
|
27.
|
|
|
|
|
Use the Commutative Property of Addition.
|
|
|
Combine like terms.
|
|
|
Since 10 is added to 17a, subtract 10 from both sides to undo the
addition.
|
|
|
|
|
|
Since a is multiplied by 17, divide both sides by 17 to undo the
multiplication.
|
|
|
|
28. 7 miles
Let d be the distance (in miles) to the movies, then
is
the number of miles after the first mile. So a formula for the total charge
could be
|
first mile charge
|
+
|
|
|
rate after first mile
|
=
|
total charge
|
|
|
4.00
|
+
|
|
|
2.75
|
=
|
20.50
|
Subtract 4.00 from each side.
|
|
|
|
|
|
2.75
|
=
|
20.50
4.00
|
|
|
|
|
|
|
2.75
|
=
|
16.5
|
Divide both sides by 2.75.
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
=
|
6
|
Add 1 to both sides.
|
|
|
|
|
|
d
|
=
|
6 + 1
|
|
|
|
|
|
|
d
|
=
|
7
|
|
29. n =
|
|
|
|
|
Combine like terms.
|
|
|
Add to undo the subtraction. Or subtract to undo the addition. Then,
divide to undo the multiplication.
|
|
n
=
|
|
30. 5 movies
Let m represent the number of movies rented each month.
Here are the costs for each company (in dollars).
To collect the variable terms on one side, subtract m from both sides.
|
7.5 m
|
=
|
2.5m m
|
|
7.5
|
=
|
1.5 m
|
Divide both sides by 1.5.
|
|
=
|
m
|
|
5
|
=
|
m
|
31. 6, 7, 8
Let g represent the greatest integer.
The expressions for the three consecutive integers from least to greatest:
,
,
g.
|
twice
|
the greatest integer
|
|
3 times
|
the least integer
|
|
|
g
|
|
|
(
)
|
|
|
To create an equation, use the additional data that 2g is 2 less
than
.
|
|
|
Solve the equation.
|
The three consecutive numbers are 6, 7, and 8.
32.
|
|
Add z to both sides.
Divide both sides by 4.
|
33. x = 25
|
|
|
|
|
Use cross products.
|
|
|
Divide both sides by 6.
|
|
|
|
34. 9 ft
|
|
Write the scale as a fraction.
|
|
|
Let x be the actual diameter.
|
|
|
Use cross products to solve.
|
|
|
|
35. 24°, 66°
Let a represent the measure of one of the complementary angles and
represent
the measure of the second angle.
|
ratio of the measures of the angles
|
is
|
4:11
|
|
|
=
|
|
Solve
.
|
|
Use cross products.
|
|
|
Distribute.
|
|
|
Add 4a to both sides.
|
|
|
Simplify.
|
|
|
Solve the equation.
|
|
|
|
|
|
Substitute 24 for a to find the measure of the second angle.
|
The measures of the complementary angles are 24° and 66°.
36. 22.4 cm
A
corresponds to L, B corresponds to M, C corresponds
to N, and D corresponds to O.
|
|
|
|
|
Use cross products.
|
|
|
Since x is multiplied by 21, divide both sides by 21 to undo the
multiplication.
|
|
|
|
|
|
|
|
_MN is 22.4 cm.
|
|
37.
;
25 feet
|
|
|
|
|
Use cross products.
|
|
|
Since x is multiplied by 7, divide both sides by 7 to undo the
multiplication.
|
|
|
|
|
|
|
|
The tree is 25 feet tall.
|
|
38. 68.75
Method 1
Use a proportion.
|
|
Use the percent proportion.
|
|
|
Let x represent the part.
|
|
|
Find the cross products.
|
|
|
Since x is multiplied by 100, divide both sides by 100 to undo
the multiplication.
|
|
|
|
|
_55% of 125 is 68.75.
|
|
Method 2
Use an equation.
|
|
Write an equation. Let x represent the part.
|
|
|
Write the percent as a decimal and multiply.
|
|
|
|
|
55% of 125 is 68.75.
|
|
39. 650%
Method 1
Use a proportion.
|
|
Use the percent proportion.
|
|
|
Let x represent the percent.
|
|
|
Find the cross products.
|
|
|
Since x is multiplied by 74, divide both sides by 74 to undo the
multiplication.
|
|
|
|
|
|
|
|
_481 is 650% of 74.
|
|
Method 2
Use an equation.
|
|
Write an equation. Let x represent the percent.
|
|
|
Since x is multiplied by 74, divide both sides by 74 to undo the
multiplication.
|
|
|
The answer is a decimal.
|
|
|
Write the decimal as a percent.
|
|
|
|
|
_481 is 650% of 74.
|
|
40. 117.86
Method 1
Use a proportion.
|
|
Use the percent proportion.
|
|
|
Let x represent the whole.
|
|
|
Find the cross products.
|
|
|
Since x is multiplied by 56, divide both sides by 56 to undo the
multiplication.
|
|
|
|
|
|
|
|
_56% of 66 is 117.86.
|
|
Method 2
Use an equation.
|
|
Write an equation. Let x represent the whole.
|
|
|
Write the percent as a decimal.
|
|
|
Since x is multiplied by 0.56, divide both sides by 0.56 to undo
the multiplication.
|
|
|
|
|
|
|
|
_56% of 66 is 117.86.
|
|
41. 5.25%
Write the formula for gross income.
gross income
(income
number
of hours)
commission
Write the formula for commission.
gross income
(income
number
of hours)
%
of total sales
|
|
Substitute value given in the problem. Let x represent the
percent commission.
|
|
|
Multiply
|
|
|
Subtract.
|
|
|
Since x is multiplied by 3,350, divide both sides by 3,350 to
undo the multiplication.
|
|
|
The answer is a decimal.
|
|
_5.25% = x
Write
the decimal as a percent.
|
|
|
|
|
_Aarons percent commission is 5.25%.
|
42. $38.75
Step 1
Find the cost of the meal before the tip and sales tax.
|
|
Write the formula for the sales tax.
|
|
|
Substitute known values.
|
|
|
Solve for c, the cost of the meal.
|
Step 2
Find the total cost of the meal,
including tip and sales tax.
|
|
Write the formula for the total cost.
|
|
|
|
|
|
Substitute the known values.
|
|
|
|
|
|
|
|
|
|
43. 200% increase
|
|
|
|
|
Substitute the given values.
|
|
|
|
If the first number is less than the second number, there is a percent of
increase.
If the first number is greater than the second number, there is a percent of
decrease.
44. m
³
3
Use the variable m. The arrow points to the right, so use either > or
³.
The solid circle at 3 means that 3 is a solution, so use
³.
45.
Step 1:
Rewrite both mixed numbers as improper fractions.
and
Step 2:
Solve the inequality.
|
|
Rewrite the inequality.
|
|
|
Subtract
from
both sides.
|
|
|
Rewrite the fractions with a common denominator.
|
|
=
|
Simplify.
|
Step 3:
Graph the inequality.
46. z
³
8
|
£
2
|
|
|
³
2(4)
|
Multiply both sides by 4 to isolate z. When you multiply by a
negative number, reverse the inequality symbol.
|
|
z
³
8
|
|
Use a solid circle when the value is included in the graph, such as with
³
or
£.
Use an empty circle when the value is not included, such as with > or <.
47.
AND
|
|
AND
|
|
Write the compound inequality using AND.
|
|
|
|
|
Solve each simple inequality.
|
|
|
|
|
Divide to undo the multiplication.
|
|
|
AND
|
|
|
First, graph the solutions of each simple inequality. Then, graph the
intersection by finding where the two graphs overlap.
48.
OR
First solve each simple inequality to obtain
OR
.
The graph of the compound inequality is the union of the graph of
and
the graph of
.
Find the union by combining the two regions.
49.
|
|
|
|
|
Add 9 to isolate the absolute-value expression.
|
|
|
Think: What numbers have an absolute value less than 8?
|
|
AND
|
is
between 8 and 8, inclusive.
|
|
AND
|
Solve the two inequalities.
|
|
|
Write the solution as a compound inequality.
|
50. y = x
+ 3
The correct equation is y = x + 3.
51. Independent: hours
worked; Dependent: total pay
The value of the dependent variable depends on the value of the
independent variable.
In this situation, the total amount Kyoko is paid depends on the number
of hours she works, so hours worked is the independent variable and total pay is
the dependent variable.
52. 6
|
|
|
|
|
Substitute 1 for x.
|
|
|
Simplify.
|
53.
Step 1:
Solve the equation for y: y =
x
+
.
Step 2:
Substitute the x-values of the given domain into the equation to find the
y-values. Make a table to help organize your work.
|
x
|
y
=
x
+
|
(x, y)
|
Step 3:
Graph the ordered pairs.
|
|
8
|
y
=
(8)
+
=
3
|
(8, 3)
|
|
4
|
y
=
(4)
+
=
1
|
(4, 1)
|
|
0
|
y
=
(0)
+
=
1
|
(0, 1)
|
|
4
|
y
=
(4)
+
=
3
|
(4, 3)
|
|
8
|
y
=
(8)
+
=
5
|
(8, 5)
|
54. Yes; common
difference 6; next 3 terms are 35, 41, 47
For a sequence to be an arithmetic sequence, each number subtracted from the one
before it should result in a common difference.
This sequence is arithmetic. Each term differs from the previous one by 6.
55. 91
Find a specific term from a given sequence by using the equation
,
where:
an
= your result
a1
= the initial term of the sequence
n
= the number in the sequence you want to calculate
d
= the common difference between the terms
n
is given in the problem, a1 is the first term in the sequence,
and d is the difference between adjacent terms.
56. 459 miles
Since you want to find the distance for 7 hours later, you need to find the 8th
term of the sequence. So, n = 8.
|
|
Write the rule to find the nth term.
|
|
an
= 60 + (8 1)(57)
|
Substitute 60 for
,
57 for d, and 8 for n.
|
|
an
= 459
|
Simplify.
|
Sylvie will have traveled 459 miles.
57. x-intercept:
8, y-intercept: 4
To find the x-intercept, replace y with 0 and solve for x;
to find the y-intercept, replace x with 0 and solve for y.
|
x-intercept
|
y-intercept
|
|
|
|
58.
|
|
Use the slope formula.
|
|
|
Substitute
for
and
for
.
|
|
=
|
Simplify.
|
59.
Find the x-intercept by substituting x = 0 into the equation. Find
the y-intercept by substituting y = 0 into the equation. Use the
two intercept points and the slope formula,
,
to calculate the slope.
60.
Plot the y-intercept 2 on the graph at (0, 2). The slope is
,
so from the y-intercept, rise 1 units and run 3 units. Plot another
point. Connect the points to graph the line.
61. y =
x
+
The slope-intercept form is y = mx + b, where m is
the slope and b is the y-intercept. Substituting
for
the slope and
for
the y-intercept gives y =
x
+
.
62. y = 4x
14
If you are given the slope and one point, you can find the y-intercept by
substituting for m, x, and y in the equation y = mx
+ b. Then, solve for b.
2 = 4(3) + b
2 = 12 + b
14 = b
So, the equation of the line in slope-intercept form is y = 4x
14.
63.
,
Plot
.
Count 1 down and 2 right, and plot another point. Draw a line connecting the two
points.
64. y + 7 =
(x
+ 8)
Substitute the point and slope into the point-slope form
,
where m represents the slope and
represents
a point on the line.
65. Lines 2 and 3 are
parallel.
Write all the equations in slope-intercept form (y = mx + b). The
equations that have the same slope but different y-intercepts are
parallel lines.
66.
ABCD
is a parallelogram because both pairs of opposite sides are parallel.
Use the ordered pairs and the slope formula to find the slopes of the four line
segments formed by the four points given. If the points given create two sets of
parallel segments, then the quadrilateral formed by the four points is a
parallelogram.
|
|
The formula for the slope between two points
(x1, y1) and (x2,
y2) is
|
67. Slope of
and
slope of
.
is
perpendicular to
because
.
LMN is a right triangle because it contains a right angle.
A right triangle contains one right angle.
L
and
N
are not right angles, so the only possibility is
M.
If LMN is a right triangle,
will
be perpendicular to
.
slope of
slope of
,
so
is
perpendicular to
.
LMN
is a right triangle because it contains a right angle.
68. yes
Substitute 5 for x and 3 for y in both equations. Since these
values make both equations true, (5, 3) is a solution of the system.
69. (
,
)
|
Step 1
|
|
The second equation is solved for y.
|
|
|
|
|
|
Step 2
|
|
|
|
|
|
Substitute
for
y in the first equation.
|
|
|
|
|
|
Step 3
|
|
Simplify and solve for x.
|
|
|
|
|
|
|
|
Divide both sides by 4.
|
|
|
x
=
|
|
|
|
|
|
|
Step 4
|
|
Write one of the original equations.
|
|
|
y =
|
Substitute
for
x.
|
|
|
|
Find the value of y.
|
|
|
|
|
|
|
(
,
)
|
Write the solution as an ordered pair.
|
70. (0, 2)
|
Step 1
|
3x 6y = 12
|
|
|
|
2x + 6y = 12
|
The y-terms have opposite coefficients.
|
|
|
5x = 0
|
Add the equations to eliminate the y terms.
|
|
|
x
= 0
|
|
|
|
|
|
|
Step 2
|
3(0) 6y = 12
|
Substitute for x in one of the original equations.
|
|
|
0 6y = 12
|
Simplify and solve for y.
|
|
|
6y = 12
|
|
|
|
y
= 2
|
|
|
|
|
|
|
|
(0, 2)
|
Write the solution as an ordered pair.
|
71. (5, 0)
|
|
Multiply all expressions in the second equation by
.
|
|
|
Add the two equations together.
|
|
|
Divide both sides by 2.
|
|
|
Solve for x.
|
|
|
|
|
|
Substitute the value for x into one of the original equations and
solve for y.
|
|
|
|
72. (4, 3)
First, multiply each equation by a number to get opposite coefficients.
|
|
|
Multiply the first equation by 3
|
|
|
|
and the second by 5 to get opposite
|
|
|
|
y-coefficients.
|
|
Step 1
|
|
Add the two equations to eliminate the y-term.
|
|
|
|
|
|
Step 2
|
|
Simplify and solve for x.
|
|
|
x
= 4
|
|
|
|
|
|
|
Step 3
|
|
Write one of the original equations.
|
|
|
|
Substitute 4 for x. Simplify and solve for y.
|
|
|
|
|
|
|
|
|
|
|
y
= 3
|
|
73. 7 zebra fish, 6
neon tetras
Let z be the number of zebra fish and let n be the number of neon
tetras that Marsha bought. Then solve the following system of equations.
|
|
Marsha spent $25.80.
Marsha bought 13 fish.
|
|
|
Multiply the second equation by 2.10
|
|
|
Add the two equations to eliminate the z term.
Solve for n.
|
To solve for z, substitute 6 for n in the first equation.
|
|
Simplify.
Solve for z.
|
74.
|
=
|
The reciprocal of 2 is
.
|
|
=
|
=
8.
|
75. 1
Any nonzero base to the zero power is equal to 1.
=
1
76.
|
|
Substitute 3 for a and 3 for b.
|
|
(
)(1)
|
Evaluate expressions with exponents.
|
|
|
Simplify.
|
77.
To multiply powers with the same base, keep the same base and add the exponents.
=
=
78.
|
|
Use the Power of a Power Property.
|
|
|
Simplify the exponent of the first term.
|
|
|
Add the exponents since the powers have the same base.
|
|
|
|
|
|
Write with a positive exponent.
|
79.
|
|
=
|
Simplify exponents with like bases:
.
|
|
|
=
|
Use the Power of a Quotient Property.
|
|
|
=
|
Use the Power of a Product Property.
|
|
|
=
|
Simplify:
.
|
|
|
=
|
Use the Power of a Power Property to simplify the exponents.
|
80.
|
=
|
Rewrite with a positive exponent.
|
|
=
=
|
Use the Power of a Quotient Property, and simplify.
|
81. 9
:
degree 9,
:
degree 2, and
:
degree 3
The degree of the polynomial is 9.
82.
|
|
= 10m + 2m4 13m
20m4
|
Identify like terms.
|
|
|
= 10m 13m + 2m4
20m4
|
Use the Commutative Property to move like terms together.
|
|
|
=
|
Combine like terms.
|
83.
|
|
|
|
=
|
Rewrite subtraction as addition of the opposite.
|
|
=
|
Identify like terms. Rearrange terms to get like terms together.
|
|
=
|
Combine like terms.
|
84.
The perimeter of a triangle is the sum of the measures of its sides. In an
isosceles triangle, the legs have equal lengths.
|
|
|
|
+
base
|
Substitute the given values.
|
|
|
Combine like terms.
|
|
|
Subtract
from
both sides.
|
|
|
Subtract
from
both sides.
|
|
|
Add 2 to both sides.
|
85.
|
|
Rearrange the terms to group like bases.
|
|
|
To multiply powers, add the exponents.
|
86.
|
|
|
|
|
Distribute
and
.
|
|
|
Distribute
and
again.
|
|
|
Multiply.
|
|
|
Combine like terms.
|
87.
|
Method 1
=
+
|
The factors fit the pattern for squaring a binomial to get a perfect
square trinomial. Use the rule for
.
|
|
|
Identify
and
from
the given binomial.
|
|
|
Use these values to determine
,
,
and
.
|
|
|
Substitute the terms into the corresponding places.
|
|
|
|
|
Method 2
|
Use FOIL to multiply the binomials.
|
88.
|
|
|
|
|
Use the rule for
.
|
|
|
Use the FOIL method, and then combine like terms.
|
|
|
Simplify.
|
89.
|
|
Find the GCF. The GCF of
,
,
and
is
|
|
|
Write the terms as products using the GCF.
|
|
|
Use the Distributive Property to factor out the GCF.
|
90.
|
|
|
|
|
Group terms that have a common number or variable as a factor.
|
|
|
Factor out the GCF of each group.
|
|
|
Factor out the common factor
.
|
91.
Find two factors of 625 such that their sum is 50. These factors are 25 and 25.
.
Simplify the expression to
.
92.
Try factors of 3 for the coefficients and factors of 8 for the constant terms.
The combination that works is:
=
=
93.
Factor out the GCF, 3z. The remaining polynomial,
,
is a perfect square trinomial that can be factored.
94. 4 and 2
|
|
|
|
|
Factor the trinomial.
|
|
or
|
Use the Zero Product Property.
|
|
or
|
Solve each equation.
|
The solutions are 4 and 2.
95.
Step 1
The area of the shaded region is equal to the area of the large rectangle minus
the area of the small rectangle.
|
|
Write an equation.
|
|
|
Use FOIL.
|
|
|
Simplify.
|
Step 2
|
|
Rewrite the equation in standard form.
|
|
|
Factor.
|
|
or
|
Use the Zero Product Property.
|
|
or
|
Solve each equation.
|
|
|
A negative number is not reasonable for distance.
|
96. There is no
solution.
Take the square root of both sides of the equation. There is no number whose
square is negative, so there is no solution.
97. The median best
describes the rents because most of the rents were near $740.
The median, or middle value, of a data set is usually the best description of
the values.
98. 99.75%
Divide the number of video players with no defects by the total number of
players. Multiply the result by 100 to express the answer as a percent.
99.
There are six possible outcomes when a fair number cube is rolled. Because the
number cube is fair, all outcomes are equally likely. There are two numbers
greater than 4 on the number cube: 5 and 6. So the probability of rolling one of
these numbers is
.
100.
,
,
Find the value of the common ratio by dividing each term by the one before it.
Then, multiply each term by the common ratio to get the next term.
101. $250
Use the equation an = a1 r n 1
to solve the problem.
The variables represent the following values:
an
= value of the computer after n years
a1
= the initial value of the computer
r
= rate of depreciation, 0.5
n
= the number of years
102.
103.
Factor perfect squares out of the radicand. Use the Product Property of Square
Roots to take the square root of each factor separately. Simplify.
104.
|
|
|
|
=
|
Use the Quotient Property.
|
|
=
|
Find perfect square factors if possible. Write 300 as 100(3).
|
|
=
|
Use the Product Property.
|
|
=
|
Simplify.
|
105.
|
|
|
|
=
|
Factor the radicands using perfect-square factors.
|
|
=
|
Use the Product Property of Square Roots.
|
|
=
|
|
|
=
|
Combine like radicals.
|
106.
+
Distribute
.
Use the Product Property of Square Roots to multiply the factors in each term.
If the radicand in either term contains any perfect square factors, factor the
radicand(s) and simplify. Combine like terms if applicable.
107.
|
|
|
Multiply by a form of 1 to get a perfect-square radicand in the
denominator.
|
|
|
|
Simplify the denominator.
|
|
|
|
|
108. z = 676
|
|
|
|
|
Add 10 to both sides.
|
|
|
Square both sides.
|
|
|
|
109.
81
There are two methods to solve the equation.
Method 1
Divide both sides by 8 and then square both sides.
Method 2
Square both sides and then divide both sides by (8)2 which is 64.
110. z = 3
|
|
|
|
|
Add
to
both sides.
|
|
|
Square both sides.
|
|
|
Simplify.
|
|
z
= 3
|
Solve.
|
111. The excluded values are 4
and 1.
The excluded values are all values of n that make the denominator equal
to zero.
|
n2
5n + 4 = 0
|
Set the denominator equal to 0.
|
|
(n 4)(n 1) = 0
|
Factor.
|
|
n
4 =0 or n 1 = 0
|
Use the Zero-Product Property to solve each factor for n.
|
|
n
= 4 or n = 1
|
4 must be excluded. 1 must be excluded.
|
The excluded values are 4 and 1.
112. 3r; r
3
Factor common factors out of the numerator and/or denominator. Divide out the
common factors to simplify the expression. Finally, use the original denominator
to determine excluded factors.
113.
|
|
|
|
=
|
Factor.
|
|
=
|
Identify the opposite binomials.
|
|
=
|
Divide out the common factors, and simplify.
|
114.
Arrange the expressions so like terms are together:
.
Multiply the numerators and denominators, remembering to add exponents when
multiplying:
.
Divide, remembering to subtract exponents:
.
Since
,
this expression simplifies to
115.
|
|
|
|
=
|
Factor the numerator and denominator.
|
|
=
|
Simplify.
|
|
=
|
Multiply the remaining factors.
|
116.
|
|
|
|
=
|
Write as multiplication by the reciprocal.
|
|
=
|
Multiply the numerators and the denominators.
|
|
=
|
Divide out common factors. Simplify.
|
117.
|
|
|
|
=
|
Factor.
|
|
=
|
Multiply the numerators and the denominators.
|
|
=
|
Divide out common factors and simplify.
|
118.
|
|
|
|
|
Combine like terms in the numerator.
|
|
|
Factor. Divide out common factors.
|
|
|
Simplify.
|
119.
|
|
Identify the LCD,
.
|
|
=
|
|
|
=
|
Rewrite each fraction with a denominator of
.
|
|
=
|
Add.
|
|
=
|
and
are
not like terms, so they cannot be combined. Factor and divide out common
factors.
|
|
=
|
Simplify.
|
120.
;
and
|
|
|
|
=
|
Factor the difference of squares.
|
|
=
|
Write each expression using the LCD.
|
|
=
|
Subtract the numerators.
|
|
=
|
Distribute.
|
|
=
|
Simplify.
|
Excluded values are values that make the denominator equal to 0. If
or
,
then the denominator is 0. Therefore,
and
.
