,OpenStax College Algebra 2e with Corequisite Support
Corequisite Skills Sections Solutions Chapter 2
2.2 Corequisite Skills: Linear Equations in One Variable
Student Learning Objectives:
LO 1) Simplify expressions using order of operations.
1. 149
2. 5
3. 31
4. 0
5a. 243
5b. 125
5c. 7
6. 0
7. 25
8. 22a+1
9. 64b+5
LO 2) Solve linear equations using a general strategy.
10. y=5
11. x=2
12. n=2
13. y=-5
14. x=1
15. m = 3.2 = 5
16
16. x=-2
17. n=9
2.3 Corequisite Skills: Models and Applications
Student Learning Objectives:
LO 1) Solve a formula for a specified variable.
1. 𝑏 = 𝑃 − 𝑎 − 𝑐
2. 𝑠 = 4
𝑃
3. 𝑟 = 2𝜋
𝐶
4. 𝑏 =
2𝐴
ℎ
5.
𝑃−2𝐿
𝑊= 2
6.
𝑦−𝑏
𝑚= 𝑥
This OpenStax ancillary resource is © 2020 Rice University.
All Rights Reserved.
,7. ℎ =
𝐴−2𝜋𝑟 2
2𝜋𝑟
8. 𝑟 = (note: a radius cannot be negative)
𝐴
𝜋
9. 𝑠 = (note: a side cannot be negative)
3𝑣
ℎ
10. 𝐿
𝐴−2𝐻𝑊
= 2𝑊+2𝐻
LO 2) Use a problem-solving strategy for word problems.
11. height=5 cm
12. Husband’s salary=$42,000
13. Protein = 50g
14. Percent increase=27.8%
15. Principle=$11,450
16. Percent tip=21.2%
2.4 Corequisite Skills: Complex Numbers
Student Learning Objectives:
LO 1) Use the product property to simplify radical expressions.
1. 3 6
2. 5 3
3. 5𝑥 𝑥
4. 5 5
3
5. 4 2
3
6. 𝑥2 𝑥2 or x^(5/2)
4
7. 2𝑟 2 4 𝑟
8. 3𝑠 2 𝑞 𝑠 2
4
LO 2) Evaluate the square root of a negative number.
9. 10i
10. 2𝑖 3
11. 5𝑖 3
12. -21
13. 54i
14. 14-8i
15. 29
16. 8+7i
17. 7-13i
18. 4
19. (25 − 5 2) + 3 2𝑖
,(Note that the textbook problems will be updated to better reflect the concepts covered in the
section. The new answers will be: 18. 3+i 19. 25 − 2𝑖 2
2.5 Corequisite Skills: Quadratic Equations
Student Learning Objectives:
LO 1) Recognize and use the appropriate method to factor a polynomial completely.
1. 𝑥 + 4 𝑥 + 6
2. 𝑦 − 18 𝑦 − 2
3. 2 𝑥 + 3 𝑥 + 5
4. 3𝑥 + 7 3𝑥 + 7
5. 5𝑛 − 9 5𝑛 − 9
6. 5xy 2𝑥 − 1 𝑥 + 7
7. 11𝑞 − 10 11𝑞 + 10
8. and 9.* 2 25𝑚 2 + 36
10. ( 5𝑧 + 3 25𝑧 2 − 15𝑧 + 9
11. 2𝑏 − 1 6𝑎 + 5
12. 2 3𝑥 − 𝑦 − 7 3𝑥 − 𝑦 + 7
13. 3𝑥 − 9 3𝑥 − 8
*Note: There was an error in the publication of the textbook, where the original version
duplicated practice problem 8, thus shifting all of the numbered items down. We will correct
this and update this manual. However, the answers as is match what is currently in the book.
This OpenStax ancillary resource is © 2020 Rice University. All Rights Reserved.
,OpenStax College Algebra 2e with Corequisite Support
Corequisite Skills Section Solutions Chapter 3
3.1 Corequisite Skills: Functions and Function Notation
LO 1) Find the value of a function. (IA 3.5.3)
Practice:
1. a. 4
b. x = 4, -4
2. a. 22
b. 17
c. 3𝑡2 − 2𝑡 + 1
d. 𝑥 = 0, 3
2
3. a. -2
b. 1
c. − 4 + 19
d. x = -1
4.a. 9
b. 5, -4
c. 0
d. No, for input of x = 9, there are two outputs
5. a. 5
b. 2
c. 4
d. x = 4, -4
e. Yes, for each input there is one output (passes vertical line test).
6. a. -22
b. -7
c. 5w+13
d. x = 6
e. 𝑥 = − 5
8
7. a. 20
b. 14
c. 4
d. 2 𝑤 − 5 + 4
e. x = 13, -3
8.a. − 2𝑥2 + 3
b. Answers will vary
,3.2 Corequisite Skills: Domain and Range
LO 1) Find the domain and range of a function. (IA 3.5.1)
Activity: Restrictions on the domain of a function.
1. division by zero is undefined.
2. taking the square root of a negative number (not in real number system).
3. − ∞, − 3 ∪ − 3, ∞ ; − 7, ∞ ; 5 ,∞ ;(− ∞, − 5) ∪ (− 5, − 2) ∪ (− 2,∞)
17
Practice:
1.a. {1,2,3,4,5}
b. {3,6,9,12,15}
2. D: [-5,1] R: [-4,2]
3. D: − ∞,∞ 𝑅: − ∞,∞
4. D: − ∞,∞ 𝑅:[2, ∞)
5. D: − ∞,∞ 𝑅:(− ∞, 3]
6. D: 3,∞ 𝑅:[0,∞)
7. D: − ∞,∞ 𝑅: − ∞,∞
8. D: − ∞, − 2 ∪ − 2 ,∞ 𝑅: − ∞, 2 ∪
1 1 1 1
2
,∞
3.3 Corequisite Skills: Rates of Change and Behaviors of Graphs
LO 1) Find the average rate of change or slope between two points on a function. (IA 3.2.1)
Activity: Finding Slopes of Lines
a. L1 and L2, m= -1/2
b. L1 and L3, or L2 and L3 ; m= -1/2 and m = 2
c. Two lines are PARALLEL if their slopes are equal.
Two lines are PERPENDICULAR if their slopes are negative reciprocals.
Practice:
1.
x y
–6 -3
–3 -1
0 1
3 3
6 5
Slope is 2/3, y intercept is (0, 1)
,2.a. 19.8
b. 140/9
c. 71
3.a.
x f(x)
-2 -8
2 0
4 16
b. 2
c. 8
4.a.
x g(x)
-1 -1/2
0 1/3
2 1/5
b. 5/6
c. 7/30
3.4 Corequisite Skills: Compositions of functions
LO 1) Find the value of a function. (IA 3.5.3)
Practice:
1.a. 3
b. 4
2.a. 3, -3
b. x = 14
3.a. 28
b. -1
c. 𝑓 𝑡 = 2𝑡2 + 3𝑡 + 1
d. x = 0, -3/2
4.a. 32
b. x = 4
5.a. 3
b. -8
c. 1
d. 3
,e. 3
f. 7
Answers for table on composite functions:
x g(x)=3x-1 f(g(x))=(g(x))2+1
-1 -4 17
-3 -10 101
0 -1 2
4 11 122
10 29 842
3.5 Corequisite Skills: Transformations of Functions
Student Learning Objectives:
LO 1) Identify graphs of basic functions. (IA 3.6.2)
Practice: Identify graphs of basic functions. (IA 3.6.2)
Problems 1-9: Table values will vary.
LO 2) Graph quadratic functions using transformations. (IA 9.7.4)
Practice: Graph quadratic functions using transformations. (IA 9.7.4)
10. Shifts the graph up and down (vertical shift).
11. Shifts the graph to the right and left (horizontal shift).
12. Widens or narrows graph ( vertical stretch or compression).
13. Reflects graph over x-axis.
14. (-3,2); (-3,-2); (3,2); (3,-2)
15.a. (-200, -67)
b. (56, 111)
16.a. up; down
b. right; left
17.a. 𝑓 𝑥 = − 𝑥 − 3 2 − 2
b. 𝑓 𝑥 = 4 𝑥 + 6 2 + 5
18.a. 𝑓 𝑥 = − 𝑥 − 2 − 1
b.𝑓 𝑥 = 3 𝑥 + 4 + 5
,3.6 Corequisite Skills: Absolute Value Functions
Student Learning Objectives:
LO 1) Solve absolute value equations (IA 2.7.1)
Practice: Solve absolute value equations (IA 2.7.1)
1. x = 0
2. x = -2, 2
3. x = 8, -4
4. x = 3, -4
5. no solution
6. z = 4, -1
7. x = 9, -3
LO 2) Identify graphs of absolute value functions (IA 3.6.2)
8.
9.
10.
, 11.
12.
13.
14. 𝐷:(− ∞, ∞) ; 𝑅:[− 4, ∞)
Corequisite Skills Sections Solutions Chapter 2
2.2 Corequisite Skills: Linear Equations in One Variable
Student Learning Objectives:
LO 1) Simplify expressions using order of operations.
1. 149
2. 5
3. 31
4. 0
5a. 243
5b. 125
5c. 7
6. 0
7. 25
8. 22a+1
9. 64b+5
LO 2) Solve linear equations using a general strategy.
10. y=5
11. x=2
12. n=2
13. y=-5
14. x=1
15. m = 3.2 = 5
16
16. x=-2
17. n=9
2.3 Corequisite Skills: Models and Applications
Student Learning Objectives:
LO 1) Solve a formula for a specified variable.
1. 𝑏 = 𝑃 − 𝑎 − 𝑐
2. 𝑠 = 4
𝑃
3. 𝑟 = 2𝜋
𝐶
4. 𝑏 =
2𝐴
ℎ
5.
𝑃−2𝐿
𝑊= 2
6.
𝑦−𝑏
𝑚= 𝑥
This OpenStax ancillary resource is © 2020 Rice University.
All Rights Reserved.
,7. ℎ =
𝐴−2𝜋𝑟 2
2𝜋𝑟
8. 𝑟 = (note: a radius cannot be negative)
𝐴
𝜋
9. 𝑠 = (note: a side cannot be negative)
3𝑣
ℎ
10. 𝐿
𝐴−2𝐻𝑊
= 2𝑊+2𝐻
LO 2) Use a problem-solving strategy for word problems.
11. height=5 cm
12. Husband’s salary=$42,000
13. Protein = 50g
14. Percent increase=27.8%
15. Principle=$11,450
16. Percent tip=21.2%
2.4 Corequisite Skills: Complex Numbers
Student Learning Objectives:
LO 1) Use the product property to simplify radical expressions.
1. 3 6
2. 5 3
3. 5𝑥 𝑥
4. 5 5
3
5. 4 2
3
6. 𝑥2 𝑥2 or x^(5/2)
4
7. 2𝑟 2 4 𝑟
8. 3𝑠 2 𝑞 𝑠 2
4
LO 2) Evaluate the square root of a negative number.
9. 10i
10. 2𝑖 3
11. 5𝑖 3
12. -21
13. 54i
14. 14-8i
15. 29
16. 8+7i
17. 7-13i
18. 4
19. (25 − 5 2) + 3 2𝑖
,(Note that the textbook problems will be updated to better reflect the concepts covered in the
section. The new answers will be: 18. 3+i 19. 25 − 2𝑖 2
2.5 Corequisite Skills: Quadratic Equations
Student Learning Objectives:
LO 1) Recognize and use the appropriate method to factor a polynomial completely.
1. 𝑥 + 4 𝑥 + 6
2. 𝑦 − 18 𝑦 − 2
3. 2 𝑥 + 3 𝑥 + 5
4. 3𝑥 + 7 3𝑥 + 7
5. 5𝑛 − 9 5𝑛 − 9
6. 5xy 2𝑥 − 1 𝑥 + 7
7. 11𝑞 − 10 11𝑞 + 10
8. and 9.* 2 25𝑚 2 + 36
10. ( 5𝑧 + 3 25𝑧 2 − 15𝑧 + 9
11. 2𝑏 − 1 6𝑎 + 5
12. 2 3𝑥 − 𝑦 − 7 3𝑥 − 𝑦 + 7
13. 3𝑥 − 9 3𝑥 − 8
*Note: There was an error in the publication of the textbook, where the original version
duplicated practice problem 8, thus shifting all of the numbered items down. We will correct
this and update this manual. However, the answers as is match what is currently in the book.
This OpenStax ancillary resource is © 2020 Rice University. All Rights Reserved.
,OpenStax College Algebra 2e with Corequisite Support
Corequisite Skills Section Solutions Chapter 3
3.1 Corequisite Skills: Functions and Function Notation
LO 1) Find the value of a function. (IA 3.5.3)
Practice:
1. a. 4
b. x = 4, -4
2. a. 22
b. 17
c. 3𝑡2 − 2𝑡 + 1
d. 𝑥 = 0, 3
2
3. a. -2
b. 1
c. − 4 + 19
d. x = -1
4.a. 9
b. 5, -4
c. 0
d. No, for input of x = 9, there are two outputs
5. a. 5
b. 2
c. 4
d. x = 4, -4
e. Yes, for each input there is one output (passes vertical line test).
6. a. -22
b. -7
c. 5w+13
d. x = 6
e. 𝑥 = − 5
8
7. a. 20
b. 14
c. 4
d. 2 𝑤 − 5 + 4
e. x = 13, -3
8.a. − 2𝑥2 + 3
b. Answers will vary
,3.2 Corequisite Skills: Domain and Range
LO 1) Find the domain and range of a function. (IA 3.5.1)
Activity: Restrictions on the domain of a function.
1. division by zero is undefined.
2. taking the square root of a negative number (not in real number system).
3. − ∞, − 3 ∪ − 3, ∞ ; − 7, ∞ ; 5 ,∞ ;(− ∞, − 5) ∪ (− 5, − 2) ∪ (− 2,∞)
17
Practice:
1.a. {1,2,3,4,5}
b. {3,6,9,12,15}
2. D: [-5,1] R: [-4,2]
3. D: − ∞,∞ 𝑅: − ∞,∞
4. D: − ∞,∞ 𝑅:[2, ∞)
5. D: − ∞,∞ 𝑅:(− ∞, 3]
6. D: 3,∞ 𝑅:[0,∞)
7. D: − ∞,∞ 𝑅: − ∞,∞
8. D: − ∞, − 2 ∪ − 2 ,∞ 𝑅: − ∞, 2 ∪
1 1 1 1
2
,∞
3.3 Corequisite Skills: Rates of Change and Behaviors of Graphs
LO 1) Find the average rate of change or slope between two points on a function. (IA 3.2.1)
Activity: Finding Slopes of Lines
a. L1 and L2, m= -1/2
b. L1 and L3, or L2 and L3 ; m= -1/2 and m = 2
c. Two lines are PARALLEL if their slopes are equal.
Two lines are PERPENDICULAR if their slopes are negative reciprocals.
Practice:
1.
x y
–6 -3
–3 -1
0 1
3 3
6 5
Slope is 2/3, y intercept is (0, 1)
,2.a. 19.8
b. 140/9
c. 71
3.a.
x f(x)
-2 -8
2 0
4 16
b. 2
c. 8
4.a.
x g(x)
-1 -1/2
0 1/3
2 1/5
b. 5/6
c. 7/30
3.4 Corequisite Skills: Compositions of functions
LO 1) Find the value of a function. (IA 3.5.3)
Practice:
1.a. 3
b. 4
2.a. 3, -3
b. x = 14
3.a. 28
b. -1
c. 𝑓 𝑡 = 2𝑡2 + 3𝑡 + 1
d. x = 0, -3/2
4.a. 32
b. x = 4
5.a. 3
b. -8
c. 1
d. 3
,e. 3
f. 7
Answers for table on composite functions:
x g(x)=3x-1 f(g(x))=(g(x))2+1
-1 -4 17
-3 -10 101
0 -1 2
4 11 122
10 29 842
3.5 Corequisite Skills: Transformations of Functions
Student Learning Objectives:
LO 1) Identify graphs of basic functions. (IA 3.6.2)
Practice: Identify graphs of basic functions. (IA 3.6.2)
Problems 1-9: Table values will vary.
LO 2) Graph quadratic functions using transformations. (IA 9.7.4)
Practice: Graph quadratic functions using transformations. (IA 9.7.4)
10. Shifts the graph up and down (vertical shift).
11. Shifts the graph to the right and left (horizontal shift).
12. Widens or narrows graph ( vertical stretch or compression).
13. Reflects graph over x-axis.
14. (-3,2); (-3,-2); (3,2); (3,-2)
15.a. (-200, -67)
b. (56, 111)
16.a. up; down
b. right; left
17.a. 𝑓 𝑥 = − 𝑥 − 3 2 − 2
b. 𝑓 𝑥 = 4 𝑥 + 6 2 + 5
18.a. 𝑓 𝑥 = − 𝑥 − 2 − 1
b.𝑓 𝑥 = 3 𝑥 + 4 + 5
,3.6 Corequisite Skills: Absolute Value Functions
Student Learning Objectives:
LO 1) Solve absolute value equations (IA 2.7.1)
Practice: Solve absolute value equations (IA 2.7.1)
1. x = 0
2. x = -2, 2
3. x = 8, -4
4. x = 3, -4
5. no solution
6. z = 4, -1
7. x = 9, -3
LO 2) Identify graphs of absolute value functions (IA 3.6.2)
8.
9.
10.
, 11.
12.
13.
14. 𝐷:(− ∞, ∞) ; 𝑅:[− 4, ∞)
