College Algebra Summer Packet
Evaluate 3𝑥 + 2 − 4𝑦 when 𝑥 = 2 and 𝑦 = 1.
Simplify: 3(2𝑥 − 7) + 𝑥
Solve 3𝑥 = 12 − 3𝑥
Solve and graph the solution: 1 + 4𝑥 ≥ 13
Solve and graph the solution: |𝑥 + 4| < 9
Solve the compound inequality. Graph the solution: 8𝑥 ≥ −48 and 2𝑥 < 10
Solve the compound inequality. Graph the solution: 3𝑥 > −6 and 𝑥 + 7 ≤ 8
Find the domain and range of the relation: {(0,3), (2,3), (5,5)}
Find the slope of the line through (0,7) and (5,8)
Find the slope of the line through (−3,2) and is a horizontal line.
Find the slope of the line through (−3,2) and is a vertical line.
𝑥+𝑦 =7
12. Solve: {
𝑥−𝑦 =3
13. A sweater costs $8 and a scarf costs $5. You pay $67 for 11 sweaters and scarves, combined.
How many of each did you buy? Be sure to define your variables, and write an algebraic system.
Solve algebraically and label your final answer.
14. Identify the vertex and the y-intercept of the graph of the function: 𝑦 = 2(𝑥 + 3)2 − 8.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
Write the number −8 + √−9 in the form of 𝑎 + 𝑏𝑖.
Simplify: (4 − 𝑖)(2 + 𝑖)
Solve the quadratic equation: 𝑥 2 + 𝑥 − 30 = 0
Solve the quadratic equation using the quadratic formula: 𝑥 2 − 5𝑥 − 36 = 0
Simplify: (7 − 4𝑖) + (3 + 𝑖)
Write a cubic polynomial function with rational coefficients in factored form whose zeros are: 1, 0 and 3.
Use either long division or synthetic division to determine whether the binomial (𝑥 + 2)is a
factor of (𝑥 3 − 𝑥 2 − 14𝑥 + 24)
Divide using long division or synthetic division: (𝑥 3 + 2𝑥 2 − 7𝑥 − 12) ÷ (𝑥 + 3)
Write the polynomial in standard form: 6 − 3𝑧 3 + 𝑧 + 5𝑧 2
Expand: (𝑥 − 2)4
Simplify each expression. Rationalize all denominators. Assume that all variables are positive.
25. √12𝑥 3 𝑦 4
3
26. 3√4𝑦 (√6𝑦 5 )
27.
6
1−√3
28.
29.
30.
31.
32.
33.
Solve √𝑥 + 6 = 𝑥. Check for extraneous roots.
Evaluate log 3 81
Write log 3𝑥 − log 4 as a single logarithm.
Solve log 3𝑥 = log 24
Solve 𝑒 𝑥 − 4 = 6. Round your answer to the nearest tenth.
Solve |2 − 𝑥| = 9
College Algebra Summer Packet
34. Solve: 2𝑦 − 7 = 8 − 3𝑦
𝑦−𝑥 =4
35. Solve the system: {
2𝑦 + 𝑥 = 8
36. Solve: 9 − 2√𝑥 = 5
37. Find the inverse of the function: 𝑓(𝑥) = −4𝑥 − 1
38. Write 4 log 2 𝑥 + 3 log 2 𝑦 as a single logarithm.
39. Write 53 = 125 in logarithmic form.
40. Write 2 ln 6 − ln 9 as a single natural logarithm.
3𝑎
𝑏
+ 3𝑎. State any restrictions on
2𝑏
3
√42 in exponential form?
41. Simplify:
42. What is
1 2
+
𝑦
43. What is 𝑥
the variables.
÷ 2𝑥𝑦?
𝑥
5⁄3
44. What is 𝑥
in radical form?
𝑥
45. Solve 27 = 9
46. A rectangular prism has a square base. The length of a side of the base is 3 less than half its
height. If the volume of the prism is 108 cubic inches, what is the length of a side of the base?
47.
48.
49.
50.
Simplify: √𝑥 2 + 8𝑥 + 16
Simplify: (𝑥 3 𝑦 −3 )2
Solve 34𝑥 + 6 = 12
Used Blue-Ray movies cost $6 each and new Blue-Ray movies cost $19 each. One day a store
sells 50 Blue-Ray movies for $638. Write a system of equations to determine the number of
each type of Blue-Ray sold. Use x for used Blue-Ray movies and y for new Blue-Ray movies.
How many of each type of movie were sold?
51. Evaluate
𝑎−𝑏
𝑎
2
if a=3 and b=-5.
52. If 𝑦 = 2𝑥 − 4𝑥 − 5, what is the value of y when x = -3?
53. If a=-2, find the value of 3(a-2)-2(a+1)
54. Evaluate
55.
56.
57.
58.
59.
𝑥𝑦−𝑦 2
2𝑥 2
if x = -2 and y = 3.
Simplify: 3a - 15b – a + 2b
Simplify: (2x-1)(4x+3)
Perform the indicated operation: (2𝑥 + 3)2
Perform the indicated operation: (𝑥 2 − 3𝑥 − 2) − (3𝑥 2 − 5𝑥 − 1)
Perform the indicated operation: 5x - 3y -(x + 4y)
60. Simplify:
2𝑦 2 +8𝑥𝑦
2𝑦
Perform the indicated operation and simplify:
61.
62.
63.
64.
(𝑥 + 2)(𝑥 2 − 3𝑥 + 1)
5(a + 2)+2(3 - a)
3x(2y - 4) - 2y(2x + 3)
3𝑥 2 𝑦(2𝑥𝑦 4 )3
College Algebra Summer Packet
65.
𝑥 −2
𝑥 −3
66. √8 + √18
2
67. (4√3)
68. 2√12 − 7√3
69.
15√12𝑥 2
3√3𝑥
Factor completely:
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
2𝑥 2 + 5𝑥 − 3
𝑥 2 + 𝑥 − 12
𝑥 2 − 4𝑦 2
1 − 16𝑦 2
𝑥 3 − 4𝑥 2 − 5𝑥
The area A of a rectangle of width W and length L is given by the formula A=LW. Write an
expression for the area of a rectangle with length twice L and width 2 units greater than W.
If A represents the number of apples purchased at 15 cents each and B represents the number
of bananas purchased at 10 cents each, write an expression for the total value of the purchase.
Al is 3 years less than twice as old as Vinnie. If x represents Vinnie’s age, write an expression for
Al’s age.
On a scale drawing, x inches represents 10 feet. How many feet does 6 inches represent?
To rent a car costs $22 per day plus 12 cents per mile for the number of miles driven. If a car is
rented for d days and driven m miles, write an expression for the total cost of the rental.
Write an expression to represent “the sum of a number x and 3 less than twice x”.
Harriet earns an 8 percent commission on her total monthly sales over $500. If her total sales
last month of d dollars was more than $500, write an expression for Harriet’s commission.
The sum of two numbers is 48. Four times the smaller number is equal to twice the larger
number. Find the two numbers.
Joan has one more than 3 times as many DVDs as Paul has. Together they have 25 DVDs. How
many DVDs does Paul have?
The price of a new stereo after adding 6 percent tax is $583. Find the cost of the stereo before
the tax.
Luis has $7.60 in dimes and quarters. If he has 40 coins in all, how many coins of each kind does
he have?
The length of a rectangle is 10 feet more than twice the width. The perimeter of the rectangle is
170 feet. Find the dimensions of the rectangle.
Sketch the line whose equation is 2x+y=5
What is the x-intercept of the line whose equation is 3x-5y=15?
Sketch the line whose equation is 2y=-4
Sketch the line whose equation is x=3
Sketch the line whose equation is 3y=x
College Algebra Summer Packet