Test Bank for Algebra for College Students, 8th Edition
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I NSTRUCTOR ’ S
R ESOURCE M ANUAL
A LGEBRA
FOR C OLLEGE S TUDENTS
EIGHTH EDITION
Robert Blitzer
Miami Dade College
R ESOURCE M ANUAL
A LGEBRA
FOR C OLLEGE S TUDENTS
EIGHTH EDITION
Robert Blitzer
Miami Dade College
Instructor’s Resource Manual with Tests
Algebra for College Students, Eighth Edition
Robert Blitzer
TABLE OF CONTENTS
MINI-LECTURES (per section) ML-1
Chapter 1 ML-1
Chapter 2 ML-11
Chapter 3 ML-19
Chapter 4 ML-25
Chapter 5 ML-32
Chapter 6 ML-43
Chapter 7 ML-52
Chapter 8 ML-61
Chapter 9 ML-67
Chapter 10 ML-75
Chapter 11 ML-83
Chapter 12 ML-88
Mini-Lectures Answers Included at end of section
ADDITIONAL EXERCISES (per section) AE-1
Chapter 1 AE-1
Chapter 2 AE-46
Chapter 3 AE-76
Chapter 4 AE-115
Chapter 5 AE-151
Chapter 6 AE-196
Chapter 7 AE-250
Chapter 8 AE-295
Chapter 9 AE-292
Chapter 10 AE-367
Chapter 11 AE-399
Chapter 12 AE-421
Additional Exercises Answers AE-457
GROUP ACTIVITIES (per chapter) A-1
Chapter 1 A-1
Chapter 2 A-2
Chapter 3 A-3
Chapter 4 A-4
Chapter 5 A-5
Chapter 6 A-6
Chapter 7 A-7
Chapter 8 A-8
Chapter 9 A-9
Chapter 10 A-10
Chapter 11 A-11
Chapter 12 A-12
Group Activities Answers A-13
Algebra for College Students, Eighth Edition
Robert Blitzer
TABLE OF CONTENTS
MINI-LECTURES (per section) ML-1
Chapter 1 ML-1
Chapter 2 ML-11
Chapter 3 ML-19
Chapter 4 ML-25
Chapter 5 ML-32
Chapter 6 ML-43
Chapter 7 ML-52
Chapter 8 ML-61
Chapter 9 ML-67
Chapter 10 ML-75
Chapter 11 ML-83
Chapter 12 ML-88
Mini-Lectures Answers Included at end of section
ADDITIONAL EXERCISES (per section) AE-1
Chapter 1 AE-1
Chapter 2 AE-46
Chapter 3 AE-76
Chapter 4 AE-115
Chapter 5 AE-151
Chapter 6 AE-196
Chapter 7 AE-250
Chapter 8 AE-295
Chapter 9 AE-292
Chapter 10 AE-367
Chapter 11 AE-399
Chapter 12 AE-421
Additional Exercises Answers AE-457
GROUP ACTIVITIES (per chapter) A-1
Chapter 1 A-1
Chapter 2 A-2
Chapter 3 A-3
Chapter 4 A-4
Chapter 5 A-5
Chapter 6 A-6
Chapter 7 A-7
Chapter 8 A-8
Chapter 9 A-9
Chapter 10 A-10
Chapter 11 A-11
Chapter 12 A-12
Group Activities Answers A-13
Instructor’s Resource Manual with Tests
Algebra for College Students, Eighth Edition
Robert Blitzer
TABLE OF CONTENTS
MINI-LECTURES (per section) ML-1
Chapter 1 ML-1
Chapter 2 ML-11
Chapter 3 ML-19
Chapter 4 ML-25
Chapter 5 ML-32
Chapter 6 ML-43
Chapter 7 ML-52
Chapter 8 ML-61
Chapter 9 ML-67
Chapter 10 ML-75
Chapter 11 ML-83
Chapter 12 ML-88
Mini-Lectures Answers Included at end of section
ADDITIONAL EXERCISES (per section) AE-1
Chapter 1 AE-1
Chapter 2 AE-46
Chapter 3 AE-76
Chapter 4 AE-115
Chapter 5 AE-151
Chapter 6 AE-196
Chapter 7 AE-250
Chapter 8 AE-295
Chapter 9 AE-292
Chapter 10 AE-367
Chapter 11 AE-399
Chapter 12 AE-421
Additional Exercises Answers AE-457
GROUP ACTIVITIES (per chapter) A-1
Chapter 1 A-1
Chapter 2 A-2
Chapter 3 A-3
Chapter 4 A-4
Chapter 5 A-5
Chapter 6 A-6
Chapter 7 A-7
Chapter 8 A-8
Chapter 9 A-9
Chapter 10 A-10
Chapter 11 A-11
Chapter 12 A-12
Group Activities Answers A-13
Algebra for College Students, Eighth Edition
Robert Blitzer
TABLE OF CONTENTS
MINI-LECTURES (per section) ML-1
Chapter 1 ML-1
Chapter 2 ML-11
Chapter 3 ML-19
Chapter 4 ML-25
Chapter 5 ML-32
Chapter 6 ML-43
Chapter 7 ML-52
Chapter 8 ML-61
Chapter 9 ML-67
Chapter 10 ML-75
Chapter 11 ML-83
Chapter 12 ML-88
Mini-Lectures Answers Included at end of section
ADDITIONAL EXERCISES (per section) AE-1
Chapter 1 AE-1
Chapter 2 AE-46
Chapter 3 AE-76
Chapter 4 AE-115
Chapter 5 AE-151
Chapter 6 AE-196
Chapter 7 AE-250
Chapter 8 AE-295
Chapter 9 AE-292
Chapter 10 AE-367
Chapter 11 AE-399
Chapter 12 AE-421
Additional Exercises Answers AE-457
GROUP ACTIVITIES (per chapter) A-1
Chapter 1 A-1
Chapter 2 A-2
Chapter 3 A-3
Chapter 4 A-4
Chapter 5 A-5
Chapter 6 A-6
Chapter 7 A-7
Chapter 8 A-8
Chapter 9 A-9
Chapter 10 A-10
Chapter 11 A-11
Chapter 12 A-12
Group Activities Answers A-13
TEST FORMS
CHAPTER 1 TESTS (6 TESTS ) T-1
Form A (FR) T-1
Form B (FR) T-3
Form C (FR) T-6
Form D (MC) T-9
Form E (MC) T-11
Form F (MC) T-14
CHAPTER 2 TESTS (6 TESTS ) T-17
Form A (FR) T-17
Form B (FR) T-21
Form C (FR) T-25
Form D (MC) T-29
Form E (MC) T-34
Form F (MC) T-39
CUMULATIVE R EVIEW 1-2 (2 TESTS ) T-44
Form A (FR) T-44
Form B (MC) T-47
CHAPTER 3 TESTS (6 TESTS ) T-51
Form A (FR) T-51
Form B (FR) T-54
Form C (FR) T-57
Form D (MC) T-60
Form E (MC) T-64
Form F (MC) T-68
CHAPTER 4 TESTS (6 TESTS ) T-72
Form A (FR) T-72
Form B (FR) T-76
Form C (FR) T-80
Form D (MC) T-84
Form E (MC) T-89
Form F (MC) T-94
CUMULATIVE R EVIEW 1-4 (2 TESTS ) T-99
Form A (FR) T-99
Form B (MC) T-102
CHAPTER 5 TESTS (6 TESTS ) T-107
Form A (FR) T-107
Form B (FR) T-110
Form C (FR) T-113
Form D (MC) T-116
Form E (MC) T-119
Form F (MC) T-122
CHAPTER 6 TESTS (6 TESTS ) T-125
Form A (FR) T-125
Form B (FR) T-127
Form C (FR) T-129
Form D (MC) T-131
Form E (MC) T-134
Form F (MC) T-137
CUMULATIVE R EVIEW 1-6 (2 TESTS ) T-140
Form A (FR) T-140
Form B (MC) T-143
CHAPTER 1 TESTS (6 TESTS ) T-1
Form A (FR) T-1
Form B (FR) T-3
Form C (FR) T-6
Form D (MC) T-9
Form E (MC) T-11
Form F (MC) T-14
CHAPTER 2 TESTS (6 TESTS ) T-17
Form A (FR) T-17
Form B (FR) T-21
Form C (FR) T-25
Form D (MC) T-29
Form E (MC) T-34
Form F (MC) T-39
CUMULATIVE R EVIEW 1-2 (2 TESTS ) T-44
Form A (FR) T-44
Form B (MC) T-47
CHAPTER 3 TESTS (6 TESTS ) T-51
Form A (FR) T-51
Form B (FR) T-54
Form C (FR) T-57
Form D (MC) T-60
Form E (MC) T-64
Form F (MC) T-68
CHAPTER 4 TESTS (6 TESTS ) T-72
Form A (FR) T-72
Form B (FR) T-76
Form C (FR) T-80
Form D (MC) T-84
Form E (MC) T-89
Form F (MC) T-94
CUMULATIVE R EVIEW 1-4 (2 TESTS ) T-99
Form A (FR) T-99
Form B (MC) T-102
CHAPTER 5 TESTS (6 TESTS ) T-107
Form A (FR) T-107
Form B (FR) T-110
Form C (FR) T-113
Form D (MC) T-116
Form E (MC) T-119
Form F (MC) T-122
CHAPTER 6 TESTS (6 TESTS ) T-125
Form A (FR) T-125
Form B (FR) T-127
Form C (FR) T-129
Form D (MC) T-131
Form E (MC) T-134
Form F (MC) T-137
CUMULATIVE R EVIEW 1-6 (2 TESTS ) T-140
Form A (FR) T-140
Form B (MC) T-143
CHAPTER 7 TESTS (6 TESTS ) T-148
Form A (FR) T-148
Form B (FR) T-151
Form C (FR) T-154
Form D (MC) T-157
Form E (MC) T-161
Form F (MC) T-164
CHAPTER 8 TESTS (6 TESTS ) T-167
Form A (FR) T-167
Form B (FR) T-171
Form C (FR) T-175
Form D (MC) T-178
Form E (MC) T-182
Form F (MC) T-186
CUMULATIVE R EVIEW 1-8 (2 TESTS ) T-190
Form A (FR) T-190
Form B (MC) T-193
CHAPTER 9 TESTS (6 TESTS ) T-197
Form A (FR) T-197
Form B (FR) T-200
Form C (FR) T-203
Form D (MC) T-206
Form E (MC) T-210
Form F (MC) T-214
CHAPTER 10 TESTS (6 T ESTS ) T-218
Form A (FR) T-218
Form B (FR) T-222
Form C (FR) T-226
Form D (MC) T-230
Form E (MC) T-235
Form F (MC) T-239
CUMULATIVE R EVIEW 1-10 (2 TESTS ) T-242
Form A (FR) T-242
Form B (MC) T-245
CHAPTER 11 TESTS (6 T ESTS ) T-249
Form A (FR) T-249
Form B (FR) T-251
Form C (FR) T-253
Form D (MC) T-255
Form E (MC) T-257
Form F (MC) T-259
CHAPTER 12 TESTS (6 T ESTS ) T-273
Form A (FR) T-273
Form B (FR) T-275
Form C (FR) T-277
Form D (MC) T-279
Form E (MC) T-281
Form F (MC) T-283
FINAL (2 TESTS ) T-286
Form A (FR) T-286
Form B (MC) T-292
Form A (FR) T-148
Form B (FR) T-151
Form C (FR) T-154
Form D (MC) T-157
Form E (MC) T-161
Form F (MC) T-164
CHAPTER 8 TESTS (6 TESTS ) T-167
Form A (FR) T-167
Form B (FR) T-171
Form C (FR) T-175
Form D (MC) T-178
Form E (MC) T-182
Form F (MC) T-186
CUMULATIVE R EVIEW 1-8 (2 TESTS ) T-190
Form A (FR) T-190
Form B (MC) T-193
CHAPTER 9 TESTS (6 TESTS ) T-197
Form A (FR) T-197
Form B (FR) T-200
Form C (FR) T-203
Form D (MC) T-206
Form E (MC) T-210
Form F (MC) T-214
CHAPTER 10 TESTS (6 T ESTS ) T-218
Form A (FR) T-218
Form B (FR) T-222
Form C (FR) T-226
Form D (MC) T-230
Form E (MC) T-235
Form F (MC) T-239
CUMULATIVE R EVIEW 1-10 (2 TESTS ) T-242
Form A (FR) T-242
Form B (MC) T-245
CHAPTER 11 TESTS (6 T ESTS ) T-249
Form A (FR) T-249
Form B (FR) T-251
Form C (FR) T-253
Form D (MC) T-255
Form E (MC) T-257
Form F (MC) T-259
CHAPTER 12 TESTS (6 T ESTS ) T-273
Form A (FR) T-273
Form B (FR) T-275
Form C (FR) T-277
Form D (MC) T-279
Form E (MC) T-281
Form F (MC) T-283
FINAL (2 TESTS ) T-286
Form A (FR) T-286
Form B (MC) T-292
TEST ANSWER KEYS T-299
Chapter 1 T-299
Chapter 2 T-301
Cumulative Review 1-2 T-305
Chapter 3 T-306
Chapter 4 T-309
Cumulative Review 1-4 T-313
Chapter 5 T-315
Chapter 6 T-317
Cumulative Review 1-6 T-318
Chapter 7 T-319
Chapter 8 T-321
Cumulative Review 1-8 T-324
Chapter 9 T-325
Chapter 10 T-327
Cumulative Review 1-10 T-331
Chapter 11 T-332
Chapter 12 T-336
Finals T-338
Chapter 1 T-299
Chapter 2 T-301
Cumulative Review 1-2 T-305
Chapter 3 T-306
Chapter 4 T-309
Cumulative Review 1-4 T-313
Chapter 5 T-315
Chapter 6 T-317
Cumulative Review 1-6 T-318
Chapter 7 T-319
Chapter 8 T-321
Cumulative Review 1-8 T-324
Chapter 9 T-325
Chapter 10 T-327
Cumulative Review 1-10 T-331
Chapter 11 T-332
Chapter 12 T-336
Finals T-338
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ML-1
Mini Lecture 1.1
Algebraic Expressions, Real Numbers and Interval Notation
Learning Objectives:
1. Translate English phrases into algebraic expressions.
2. Evaluate algebraic expressions.
3. Use mathematical models.
4. Recognize the sets that make up the real numbers.
5. Use set-builder notation.
6. Use the symbols ∈ and ∉.
7. Use inequality symbols.
8. Use interval notation.
Examples:
1. Write each English phrase as an algebraic expression. Let x represent the number.
a. Three less than five times a number.
b. The product of a number and six, increased by four.
2. Evaluate each algebraic expression for the given value or values of the variable(s).
a. 352 ++ xx , for x = 2
b. )(22 yxx ++ , for x = 3 y = 4
3. Use the roster method to list the elements in each set.
a. {x | x is an integer between 4 and 9}
b. {x | x is an even whole number less than 10}
4. Use the meaning of the symbols ∈ and ∉ to determine whether each statement is true or
false.
a. 3∈{x | x is a natural number}
b. 9 ∉ {1, 3, 5, 7}
5. Write out the meaning of each inequality. Then determine whether the inequality is true
or false.
a. –10 > –8 b. 02– ≤ c. 33 ≥ d. 5–2 ≤
6. Express the interval [ )5,- ¥ in set builder notation and graph.
Teaching Notes:
• Be sure to go over important vocabulary for the section including: variable, algebraic
expression, constant, exponential expression, equation, formula, natural numbers, whole
numbers, integers, rational, irrational and real numbers.
• Brainstorm the many words that translate to the four basic operations. Ex: increased -
addition.
• n
b = b· b· … b (b appears as a factor “n” times).
• Order of operation rules include:
1. First, perform all operations within grouping symbols.
2. Evaluate all exponential expressions.
3. Do all multiplication and division in the order in which they occur, working from left
to right.
4. Last, do all additions and subtractions in the order in which they occur, working from
left to right.
• < is read “less than”, > is read “greater than”
• ≤ is read “less than or equal to”, ≥ is read “greater than or equal to”
Mini Lecture 1.1
Algebraic Expressions, Real Numbers and Interval Notation
Learning Objectives:
1. Translate English phrases into algebraic expressions.
2. Evaluate algebraic expressions.
3. Use mathematical models.
4. Recognize the sets that make up the real numbers.
5. Use set-builder notation.
6. Use the symbols ∈ and ∉.
7. Use inequality symbols.
8. Use interval notation.
Examples:
1. Write each English phrase as an algebraic expression. Let x represent the number.
a. Three less than five times a number.
b. The product of a number and six, increased by four.
2. Evaluate each algebraic expression for the given value or values of the variable(s).
a. 352 ++ xx , for x = 2
b. )(22 yxx ++ , for x = 3 y = 4
3. Use the roster method to list the elements in each set.
a. {x | x is an integer between 4 and 9}
b. {x | x is an even whole number less than 10}
4. Use the meaning of the symbols ∈ and ∉ to determine whether each statement is true or
false.
a. 3∈{x | x is a natural number}
b. 9 ∉ {1, 3, 5, 7}
5. Write out the meaning of each inequality. Then determine whether the inequality is true
or false.
a. –10 > –8 b. 02– ≤ c. 33 ≥ d. 5–2 ≤
6. Express the interval [ )5,- ¥ in set builder notation and graph.
Teaching Notes:
• Be sure to go over important vocabulary for the section including: variable, algebraic
expression, constant, exponential expression, equation, formula, natural numbers, whole
numbers, integers, rational, irrational and real numbers.
• Brainstorm the many words that translate to the four basic operations. Ex: increased -
addition.
• n
b = b· b· … b (b appears as a factor “n” times).
• Order of operation rules include:
1. First, perform all operations within grouping symbols.
2. Evaluate all exponential expressions.
3. Do all multiplication and division in the order in which they occur, working from left
to right.
4. Last, do all additions and subtractions in the order in which they occur, working from
left to right.
• < is read “less than”, > is read “greater than”
• ≤ is read “less than or equal to”, ≥ is read “greater than or equal to”
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ML-2
Answers: 1. a. 3–5x b. 46 +x 2. a. 17 b. 23 3. a. {5, 6, 7, 8} b. {0, 2, 4, 6, 8}
4. a. true b. true 5. a.–10 greater than –8, false b.–2 is less than or equal to 0, true
c. 3 is greater than or equal to 3, true d. 2 is less than or equal to –5, false 6. { }| 5x x ³ -
Answers: 1. a. 3–5x b. 46 +x 2. a. 17 b. 23 3. a. {5, 6, 7, 8} b. {0, 2, 4, 6, 8}
4. a. true b. true 5. a.–10 greater than –8, false b.–2 is less than or equal to 0, true
c. 3 is greater than or equal to 3, true d. 2 is less than or equal to –5, false 6. { }| 5x x ³ -
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ML-3
Mini Lecture 1.2
Operations With Real Numbers and Simplifying Algebraic Expressions
Learning Objectives:
1. Find a number’s absolute value.
2. Add real numbers.
3. Find opposites.
4. Subtract real numbers.
5. Multiply real numbers.
6. Evaluate exponential expressions.
7. Divide real numbers.
8. Use the order of operations.
9. Use commutative, associative, and distributive properties.
10. Simplify algebraic expressions.
Examples:
1. Find the absolute value.
a. 8– b. 4
3
– c. 24.6– d. 12
2. Add or subtract.
a. –14 + 25 b. 3
1
–
4
3
– + c. 15 – (–10) d. 6.8 – 12.32
e. 8
3
–
8
5
– f. –52 + 52 g. –32 – (–38) h. 4.2 – (–8.1)
3. Evaluate.
a. ( )2
8– b. 2
8– c. ( )4
3– d. 4
3–
4. Multiply or divide.
a. 20
8
5
3
– ÷ b. (15) (–1) (–4) c. 0
24
– d. (–3.3) (1.2)
e. ( )( )( )( )2076 − f. 18
0 g. 3
2
8 −
÷− h. 15
14
7
3 −
⋅
5. Use the distributive property and simplify.
a. 6(x –2) b. –3 (6 – y) c. –4(x – 5 – y)
6. Rewrite to show how the associative property could be used to simplify the expression.
Then simplify.
a. 6(–4x) b. (x + 124) + 376
7. Simplify using the order of operation.
a. 64205 +÷⋅ b. 2
2–15
)2(5–)2(–6 c. ( )6 3 2 3x x- -
Mini Lecture 1.2
Operations With Real Numbers and Simplifying Algebraic Expressions
Learning Objectives:
1. Find a number’s absolute value.
2. Add real numbers.
3. Find opposites.
4. Subtract real numbers.
5. Multiply real numbers.
6. Evaluate exponential expressions.
7. Divide real numbers.
8. Use the order of operations.
9. Use commutative, associative, and distributive properties.
10. Simplify algebraic expressions.
Examples:
1. Find the absolute value.
a. 8– b. 4
3
– c. 24.6– d. 12
2. Add or subtract.
a. –14 + 25 b. 3
1
–
4
3
– + c. 15 – (–10) d. 6.8 – 12.32
e. 8
3
–
8
5
– f. –52 + 52 g. –32 – (–38) h. 4.2 – (–8.1)
3. Evaluate.
a. ( )2
8– b. 2
8– c. ( )4
3– d. 4
3–
4. Multiply or divide.
a. 20
8
5
3
– ÷ b. (15) (–1) (–4) c. 0
24
– d. (–3.3) (1.2)
e. ( )( )( )( )2076 − f. 18
0 g. 3
2
8 −
÷− h. 15
14
7
3 −
⋅
5. Use the distributive property and simplify.
a. 6(x –2) b. –3 (6 – y) c. –4(x – 5 – y)
6. Rewrite to show how the associative property could be used to simplify the expression.
Then simplify.
a. 6(–4x) b. (x + 124) + 376
7. Simplify using the order of operation.
a. 64205 +÷⋅ b. 2
2–15
)2(5–)2(–6 c. ( )6 3 2 3x x- -
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ML-4
Teaching Notes:
• Remind students that absolute value measures distance from zero, and for that reason, it
is always positive.
• Opposites and additives inverses are just different names for the same thing.
• Students need to be reminded often that a negative sign is only part of the base if it is
inside parentheses with the base.
• When opposites are added, the result is zero.
• Make sure students understand what is behind subtraction – why subtraction can be
changed to addition of the opposite.
• Never, never, never multiply the base and the exponent together! Students are often
tempted to do this.
Answers: 1. a. 8 b. 4
3 c. 6.24 d. 12 2. a. 11 b. 12
1
1–
12
13
– or c. 25 d. –5.52 e. –1 f. 0
g. 6 h. 12.3 3. a. 64 b. –64 c. 81 d. –81 4. a. 2
3
– b. 60 c. undefined d. –3.96 e. 0 f. 0 g. 12
h. 5
2− 5. a. 6x – 12 b. –18 + 3y c. –4x + 20 + 4y 6. a. xx 24–)4–6( =⋅
b. 500)376124( +=++ xx 7. a. 31 b. –2 c. 15x – 12
Teaching Notes:
• Remind students that absolute value measures distance from zero, and for that reason, it
is always positive.
• Opposites and additives inverses are just different names for the same thing.
• Students need to be reminded often that a negative sign is only part of the base if it is
inside parentheses with the base.
• When opposites are added, the result is zero.
• Make sure students understand what is behind subtraction – why subtraction can be
changed to addition of the opposite.
• Never, never, never multiply the base and the exponent together! Students are often
tempted to do this.
Answers: 1. a. 8 b. 4
3 c. 6.24 d. 12 2. a. 11 b. 12
1
1–
12
13
– or c. 25 d. –5.52 e. –1 f. 0
g. 6 h. 12.3 3. a. 64 b. –64 c. 81 d. –81 4. a. 2
3
– b. 60 c. undefined d. –3.96 e. 0 f. 0 g. 12
h. 5
2− 5. a. 6x – 12 b. –18 + 3y c. –4x + 20 + 4y 6. a. xx 24–)4–6( =⋅
b. 500)376124( +=++ xx 7. a. 31 b. –2 c. 15x – 12
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ML-5
Mini Lecture 1.3
Graphing Equations
Learning Objectives:
1. Plot points in the rectangular coordinate system.
2. Graph equations in the rectangular coordinate system.
3. Use the rectangular system to visualize relationships between variables.
4. Interpret information about a graphing utility’s viewing rectangle or table.
Examples:
1. Plot the following point in a rectangular coordinate system.
A. (–2, 3) B. (–4, 0) C. (1, 5)
D. (–1, –4) E. (3, –3) F. (0, 2)
2. Complete the table of values for 3–xy = , then graph the equation.
3. Complete the table of values for 2
–2 xy = , then graph the equation.
4. Complete the table of values for ,1–xy = then graph the equation.
x y = |x –1| (x, y)
–4
–3
–2
–1
0
1
2
3
4
x y = x – 3 (x, y)
–2
–1
0
1
2
x y = 2 – x2 (x, y)
–3
–2
–1
0
1
2
3
Mini Lecture 1.3
Graphing Equations
Learning Objectives:
1. Plot points in the rectangular coordinate system.
2. Graph equations in the rectangular coordinate system.
3. Use the rectangular system to visualize relationships between variables.
4. Interpret information about a graphing utility’s viewing rectangle or table.
Examples:
1. Plot the following point in a rectangular coordinate system.
A. (–2, 3) B. (–4, 0) C. (1, 5)
D. (–1, –4) E. (3, –3) F. (0, 2)
2. Complete the table of values for 3–xy = , then graph the equation.
3. Complete the table of values for 2
–2 xy = , then graph the equation.
4. Complete the table of values for ,1–xy = then graph the equation.
x y = |x –1| (x, y)
–4
–3
–2
–1
0
1
2
3
4
x y = x – 3 (x, y)
–2
–1
0
1
2
x y = 2 – x2 (x, y)
–3
–2
–1
0
1
2
3
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ML-6
Teaching Notes:
• The point plotting method is one method for graphing equations.
• The graph of a linear equation is a line.
• The graph of a quadratic equation is a parabola.
• The graph of an absolute value equation is a “V” shape that can shoot upward or
downward.
Answers: 1.
2. (–2, –5) (–1, –4) (0, –3) (1, –2) (2, –1)
3. (–3, –7) (–2, –2) (–1, 1) (0, 2) (1, 1) (2, –2) (3, –7)
4. (–4, 5) (–3, 4) (–2, 3) (–1, 2) (0, 1) (1, 0) (2, 1) (3, 2) (4, 3)
Figure for Answer 3
Figure for Answer 4
Teaching Notes:
• The point plotting method is one method for graphing equations.
• The graph of a linear equation is a line.
• The graph of a quadratic equation is a parabola.
• The graph of an absolute value equation is a “V” shape that can shoot upward or
downward.
Answers: 1.
2. (–2, –5) (–1, –4) (0, –3) (1, –2) (2, –1)
3. (–3, –7) (–2, –2) (–1, 1) (0, 2) (1, 1) (2, –2) (3, –7)
4. (–4, 5) (–3, 4) (–2, 3) (–1, 2) (0, 1) (1, 0) (2, 1) (3, 2) (4, 3)
Figure for Answer 3
Figure for Answer 4
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ML-7
Mini Lecture 1.4
Solving Linear Equations
Learning Objectives:
1. Solve linear equations.
2. Recognize identities, conditional equations, and inconsistent equations.
3. Solve applied problems using mathematical models.
Examples:
Solve each equation. If fractions are involved, you may want to clear the fractions first.
1. a. 2275 =+x b. xx 7332 =+ c. 9–23 =+ x
2. a. )1–2(4)23(5 xx =+ b. )4–(5)6–2(–)4–(5 xxx = c. )5–(3–25 xx =
3. a. 10
7–
5
4 xx = b. 6
5
3
–2
–
8
1 =
+ aa
c. 1
5
1–3
–
3
4– =
yy d. )5–3(
5
1
)82(
3
1 xx =+
Solve and determine whether the equation is an identity, a conditional equation, or an
inconsistent equation.
4. a. xxx 3)4–(235 +=+ b. )4(488)4–2(6 ++=+ xxx
c. 3)1(5)4(23 ++=++ aaa d. )33(8)44(6 +=+ yy
e. 14–4.110–6.0 xx = f. 0)–2(3)1–(6 =+ xx
Teaching Notes:
• Students may need to be reminded there is no “right” or “wrong” side of the equation.
Some students have a problem when the variable ends up on the right side of the
equation.
• Students need to practice clearing equations of fractions by multiplying each term
(whether it is a fraction or not) by the least common denominator of all the terms.
Answers:
1. a. 3 b. 8 c. –6 2. a. –2 b. 3 c. 10 3. a. 10 b. 3 c. –8 d. –55 4. a. Inconsistent ; No
Solution b. Inconsistent; No Solution c. Identity; infinitely many solutions d. Identity; infinitely many
solutions e. 5; conditional equation f. 0; conditional equation
Mini Lecture 1.4
Solving Linear Equations
Learning Objectives:
1. Solve linear equations.
2. Recognize identities, conditional equations, and inconsistent equations.
3. Solve applied problems using mathematical models.
Examples:
Solve each equation. If fractions are involved, you may want to clear the fractions first.
1. a. 2275 =+x b. xx 7332 =+ c. 9–23 =+ x
2. a. )1–2(4)23(5 xx =+ b. )4–(5)6–2(–)4–(5 xxx = c. )5–(3–25 xx =
3. a. 10
7–
5
4 xx = b. 6
5
3
–2
–
8
1 =
+ aa
c. 1
5
1–3
–
3
4– =
yy d. )5–3(
5
1
)82(
3
1 xx =+
Solve and determine whether the equation is an identity, a conditional equation, or an
inconsistent equation.
4. a. xxx 3)4–(235 +=+ b. )4(488)4–2(6 ++=+ xxx
c. 3)1(5)4(23 ++=++ aaa d. )33(8)44(6 +=+ yy
e. 14–4.110–6.0 xx = f. 0)–2(3)1–(6 =+ xx
Teaching Notes:
• Students may need to be reminded there is no “right” or “wrong” side of the equation.
Some students have a problem when the variable ends up on the right side of the
equation.
• Students need to practice clearing equations of fractions by multiplying each term
(whether it is a fraction or not) by the least common denominator of all the terms.
Answers:
1. a. 3 b. 8 c. –6 2. a. –2 b. 3 c. 10 3. a. 10 b. 3 c. –8 d. –55 4. a. Inconsistent ; No
Solution b. Inconsistent; No Solution c. Identity; infinitely many solutions d. Identity; infinitely many
solutions e. 5; conditional equation f. 0; conditional equation
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ML-8
Mini Lecture 1.5
Problem Solving and Using Formulas
Learning Objectives:
1. Solve algebraic word problems using linear equations.
2. Solve a formula for a variable.
Examples:
Solve the following using the five step strategy for solving word problems.
1. When 12 is subtracted from three times a number, the result is 36. What is the
number?
2. 15% of what number is 255?
3. In a triangle, the measure of the third angle is twice the measure of the first angle.
The measure of the second angle is twenty more than the first. Find the measure of
each angle.
4. The dog run is six feet longer than it is wide and the perimeter measures 32 feet.
Determine the measurements of the length and width of the dog run.
5. A new automobile sells for $28,000. If the mark-up is 25% of the dealer’s cost, what
is the dealer’s cost?
Solve each formula for the specified variable.
6. BhV 3
1
= for B 7. )( rtlPA += for P
8. )2–(180 nS = for n 9. 2
d
KMm
f = for M
10. CByAx =+ for A
Teaching Notes:
• Use the five step strategy for solving word problems.
• Read the problem carefully. Let a variable represent one of the quantities in the problem.
• If necessary, write an expression for any other unknown quantities in the problem in
terms of the same variable used in step 1.
• Write an equation to describe the conditions of the problem.
• Solve the equation and answer the problem’s question.
• Remind students to always check to make sure their answer makes sense.
Answers: 1. 16;3612–3 == xx 2. 1700;25515.0 == xx 3.
80,60,40;1802)20( =+++ xxx 4. longfeet116x,feet wide5;32)6(22 =+==++ xxx
5. 400,22$;000,2825.0 ==+ xxx 6. h
V
B 3
= 7. rtl
A
P +
= 8. 2
180 += S
n
9. Km
fd
M
2
= 10. x
ByC
A –
=
Mini Lecture 1.5
Problem Solving and Using Formulas
Learning Objectives:
1. Solve algebraic word problems using linear equations.
2. Solve a formula for a variable.
Examples:
Solve the following using the five step strategy for solving word problems.
1. When 12 is subtracted from three times a number, the result is 36. What is the
number?
2. 15% of what number is 255?
3. In a triangle, the measure of the third angle is twice the measure of the first angle.
The measure of the second angle is twenty more than the first. Find the measure of
each angle.
4. The dog run is six feet longer than it is wide and the perimeter measures 32 feet.
Determine the measurements of the length and width of the dog run.
5. A new automobile sells for $28,000. If the mark-up is 25% of the dealer’s cost, what
is the dealer’s cost?
Solve each formula for the specified variable.
6. BhV 3
1
= for B 7. )( rtlPA += for P
8. )2–(180 nS = for n 9. 2
d
KMm
f = for M
10. CByAx =+ for A
Teaching Notes:
• Use the five step strategy for solving word problems.
• Read the problem carefully. Let a variable represent one of the quantities in the problem.
• If necessary, write an expression for any other unknown quantities in the problem in
terms of the same variable used in step 1.
• Write an equation to describe the conditions of the problem.
• Solve the equation and answer the problem’s question.
• Remind students to always check to make sure their answer makes sense.
Answers: 1. 16;3612–3 == xx 2. 1700;25515.0 == xx 3.
80,60,40;1802)20( =+++ xxx 4. longfeet116x,feet wide5;32)6(22 =+==++ xxx
5. 400,22$;000,2825.0 ==+ xxx 6. h
V
B 3
= 7. rtl
A
P +
= 8. 2
180 += S
n
9. Km
fd
M
2
= 10. x
ByC
A –
=
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ML-9
Mini Lecture 1.6
Properties of Integral Exponents
Learning Objectives:
1. Use the product rule.
2. Use the quotient rule.
3. Use the zero-exponent rule.
4. Use the negative exponent rule.
5. Use the power rule.
6. Find the power of a product.
7. Find the power of a quotient.
8. Simplify exponential expressions.
Examples:
Simplify. Final answers should not contain any negative exponents.
1. a. 34 yy ⋅ b. )3)(4( 4
aa c. )4)(–( 34 xyzzxy d. )6)(( 3–353
2
1 nmnm
2. a. 4
10
y
y b. 2
6
5
25
a
a c. 6–
3
x
x d. 2–42
64
6
18
zyx
zyx
3. a. 2–
4 b. 3–
5
1 c. 0
6 d. 0
6x
e. 3–2–
3 yx f. 5–4
7 yx g. 8–
2–
a
a h. 1–
15
4. a. 210 )(x b. 3–6– )( y c. 1–2 )4( d. 24 )(a
5. a. 202 )3( ba b. 2–2–3– )5( yx c.
2–
3
2
d.
2
3–
2
2
6
y
x
6. a. 223–6 )4)(5( yxyx b. 2–44–
2–2–3
)6(
)2(
yx
yx c. 2–3–
34
)2(
5
yx
yx d.
3–
64
1
ba
Teaching Notes:
• Exponent rules are very easy as presented – one at a time. Students often become
confused when several rules are used in one problem. Constant reinforcement and lots of
practice will help.
• Remind students that when a variable appears to have no exponent – there is an invisible
exponent of one.
• Never, never, never multiply a base and an exponent together.
• Always (exception: scientific notation) write final answers with positive exponents only.
Answers: 1. a. 7
y b. 5
12a c. 452
4– zyx d. 26
3
Mini Lecture 1.6
Properties of Integral Exponents
Learning Objectives:
1. Use the product rule.
2. Use the quotient rule.
3. Use the zero-exponent rule.
4. Use the negative exponent rule.
5. Use the power rule.
6. Find the power of a product.
7. Find the power of a quotient.
8. Simplify exponential expressions.
Examples:
Simplify. Final answers should not contain any negative exponents.
1. a. 34 yy ⋅ b. )3)(4( 4
aa c. )4)(–( 34 xyzzxy d. )6)(( 3–353
2
1 nmnm
2. a. 4
10
y
y b. 2
6
5
25
a
a c. 6–
3
x
x d. 2–42
64
6
18
zyx
zyx
3. a. 2–
4 b. 3–
5
1 c. 0
6 d. 0
6x
e. 3–2–
3 yx f. 5–4
7 yx g. 8–
2–
a
a h. 1–
15
4. a. 210 )(x b. 3–6– )( y c. 1–2 )4( d. 24 )(a
5. a. 202 )3( ba b. 2–2–3– )5( yx c.
2–
3
2
d.
2
3–
2
2
6
y
x
6. a. 223–6 )4)(5( yxyx b. 2–44–
2–2–3
)6(
)2(
yx
yx c. 2–3–
34
)2(
5
yx
yx d.
3–
64
1
ba
Teaching Notes:
• Exponent rules are very easy as presented – one at a time. Students often become
confused when several rules are used in one problem. Constant reinforcement and lots of
practice will help.
• Remind students that when a variable appears to have no exponent – there is an invisible
exponent of one.
• Never, never, never multiply a base and an exponent together.
• Always (exception: scientific notation) write final answers with positive exponents only.
Answers: 1. a. 7
y b. 5
12a c. 452
4– zyx d. 26
3
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ML-10
Mini Lecture 1.7
Scientific Notation
Learning Objectives:
1. Convert from scientific to decimal notation.
2. Convert from decimal to scientific notation.
3. Perform computations with scientific notation.
4. Use scientific notation to solve problems.
Examples:
1. Write each number in decimal notation.
a. 5
104.3– × b. 4–
10015.2 ×
2. Write each number in scientific notation.
a. 32,500,000,000 b. –0.00417 c. 4
109432 ×
3. Perform the indicated computations, writing the answers in scientific notation.
a. )108)(104.2( 5–3 ×× b. 2–
4
104
108.6
×
×
4. In Central City, the population is 176,000. Express the population in scientific notation.
Teaching Notes:
• A number is written in scientific notation when it is expressed in the form n
a 10× with
1 ≤ | a | < 10 and “n” is an integer.
• When multiplying terms written in scientific notation mnmn baba +
××=×× 10)()10)(10( .
• When dividing terms written in scientific notation .10
10
10 – nm
n
m
b
a
b
a ×=
×
×
• When multiplying or dividing is complete, make sure the final answer is in scientific
notation.
• Students need to be reminded that a number must be written as a number between 1 and
10 to be in scientific notation.
• The sign of a number has nothing to do with the sign of the power when a number is
written in scientific notation.
Answers: 1. a. –340,000 b. 0.0002015 2. a. 10
1025.3 × b. 3–
1017.4 × c. 7
10432.9 ×
3. a. 1–
1092.1 × b. 6
107.1 × 4. 5
1076.1 ×
Mini Lecture 1.7
Scientific Notation
Learning Objectives:
1. Convert from scientific to decimal notation.
2. Convert from decimal to scientific notation.
3. Perform computations with scientific notation.
4. Use scientific notation to solve problems.
Examples:
1. Write each number in decimal notation.
a. 5
104.3– × b. 4–
10015.2 ×
2. Write each number in scientific notation.
a. 32,500,000,000 b. –0.00417 c. 4
109432 ×
3. Perform the indicated computations, writing the answers in scientific notation.
a. )108)(104.2( 5–3 ×× b. 2–
4
104
108.6
×
×
4. In Central City, the population is 176,000. Express the population in scientific notation.
Teaching Notes:
• A number is written in scientific notation when it is expressed in the form n
a 10× with
1 ≤ | a | < 10 and “n” is an integer.
• When multiplying terms written in scientific notation mnmn baba +
××=×× 10)()10)(10( .
• When dividing terms written in scientific notation .10
10
10 – nm
n
m
b
a
b
a ×=
×
×
• When multiplying or dividing is complete, make sure the final answer is in scientific
notation.
• Students need to be reminded that a number must be written as a number between 1 and
10 to be in scientific notation.
• The sign of a number has nothing to do with the sign of the power when a number is
written in scientific notation.
Answers: 1. a. –340,000 b. 0.0002015 2. a. 10
1025.3 × b. 3–
1017.4 × c. 7
10432.9 ×
3. a. 1–
1092.1 × b. 6
107.1 × 4. 5
1076.1 ×
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ML-11
Mini Lecture 2.1
Introduction to Functions
Learning Objectives:
1. Find the domain and range of a relation.
2. Determine whether a relation is a function.
3. Evaluate a function.
Examples:
1. Find the domain and range of the relation.
a. {(1, 5), (2, 10), (3, 15), (4, 20), (5, 25)} b. {(1, -1), (0, 0), (-5, 5)}
2. Determine whether each relation is a function.
a. { })10,5(),9,5(),8,5(),7,5(),6,5( b. {(5, 6), (6, 7), (7, 8), (8, 9), (9, 10)}
3. Find the indicated function value.
a. 2–3)(for)3( xxff = b. 4–2)(for)2(– 2 += xxxgg
c. 23–)(for)1(– 2 += ttthh d. 32)(for)( +=+ xxfhaf
4. Function g is defined by the table
Find the indicated function value.
a. g(2) b. g (4)
Teaching Notes:
• A relation is any set of ordered pairs.
• The set of all first terms “x-values” of the ordered pairs is called the domain.
• The set of all second terms “y-values” of the ordered pairs is called the range.
• A function is a relation in which each member of the domain corresponds to exactly one
member of the range.
• A function is a relation in which no two ordered pairs have the same first component and
different second components.
• The variable “x” is called the independent variable because it can be assigned any value
from the domain.
• The variable “y” is called the dependent variable because its value depends on “x”.
• The notation f(x), read “f of x” represents the value of the function at the number “x”.
Answers: 1. domain {1, 0, -5} range {-1, 0, 5} 2. a. not a function b. function
3. a. 7 b. 14 c. 6 d. 322 ++ ha n 4. a. 6 b. 10
x g (x)
0 2
1 4
2 6
3 8
4 10
Mini Lecture 2.1
Introduction to Functions
Learning Objectives:
1. Find the domain and range of a relation.
2. Determine whether a relation is a function.
3. Evaluate a function.
Examples:
1. Find the domain and range of the relation.
a. {(1, 5), (2, 10), (3, 15), (4, 20), (5, 25)} b. {(1, -1), (0, 0), (-5, 5)}
2. Determine whether each relation is a function.
a. { })10,5(),9,5(),8,5(),7,5(),6,5( b. {(5, 6), (6, 7), (7, 8), (8, 9), (9, 10)}
3. Find the indicated function value.
a. 2–3)(for)3( xxff = b. 4–2)(for)2(– 2 += xxxgg
c. 23–)(for)1(– 2 += ttthh d. 32)(for)( +=+ xxfhaf
4. Function g is defined by the table
Find the indicated function value.
a. g(2) b. g (4)
Teaching Notes:
• A relation is any set of ordered pairs.
• The set of all first terms “x-values” of the ordered pairs is called the domain.
• The set of all second terms “y-values” of the ordered pairs is called the range.
• A function is a relation in which each member of the domain corresponds to exactly one
member of the range.
• A function is a relation in which no two ordered pairs have the same first component and
different second components.
• The variable “x” is called the independent variable because it can be assigned any value
from the domain.
• The variable “y” is called the dependent variable because its value depends on “x”.
• The notation f(x), read “f of x” represents the value of the function at the number “x”.
Answers: 1. domain {1, 0, -5} range {-1, 0, 5} 2. a. not a function b. function
3. a. 7 b. 14 c. 6 d. 322 ++ ha n 4. a. 6 b. 10
x g (x)
0 2
1 4
2 6
3 8
4 10
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ML-12
X
Y
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
0
Mini Lecture 2.2
Graphs of Functions
Learning Objectives:
1. Graph functions by plotting points.
2. Use the vertical line test to identify functions.
3. Obtain information about a function from its graph.
4. Identify the domain and range of a function from its graph.
Examples:
State the domain of each function.
1. Graph the function 13)(and3)( +== xxgxxf in the same rectangular coordinate
system. Graph integers for x starting with –2 and ending with 2. How is the graph of g
related to the graph of f ?
2. Use the vertical line test to identify graphs in which y is a function of x.
a. b. c.
3. Use the graph of f to find the indicated function value.
a. f(2) b. f(0) c. f(1)
4. Use the graph each function to identify its domain and range.
a. b.
X
Y
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
0
(-5 ,1) (1, 1) (-2, 1) (4, 1)
X
Y
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
0
X
Y
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
0
Mini Lecture 2.2
Graphs of Functions
Learning Objectives:
1. Graph functions by plotting points.
2. Use the vertical line test to identify functions.
3. Obtain information about a function from its graph.
4. Identify the domain and range of a function from its graph.
Examples:
State the domain of each function.
1. Graph the function 13)(and3)( +== xxgxxf in the same rectangular coordinate
system. Graph integers for x starting with –2 and ending with 2. How is the graph of g
related to the graph of f ?
2. Use the vertical line test to identify graphs in which y is a function of x.
a. b. c.
3. Use the graph of f to find the indicated function value.
a. f(2) b. f(0) c. f(1)
4. Use the graph each function to identify its domain and range.
a. b.
X
Y
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
0
(-5 ,1) (1, 1) (-2, 1) (4, 1)
X
Y
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
0
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ML-13
Teaching notes:
• The graph of a function is the graph of the ordered pairs.
• If a vertical line intersects a graph in more than one point, the graph does not define y
as a function of x.
Answers:
1. The graph of g is the graph of f shifted up 1 unit. 2. a. yes b. no c. yes 3. a. 0 b. 4 c. 1
4. a. Domain: { }5,1, 1.4- Range: { }1 b. Domain: [ )0, ¥ Range: [ )2, ¥
Teaching notes:
• The graph of a function is the graph of the ordered pairs.
• If a vertical line intersects a graph in more than one point, the graph does not define y
as a function of x.
Answers:
1. The graph of g is the graph of f shifted up 1 unit. 2. a. yes b. no c. yes 3. a. 0 b. 4 c. 1
4. a. Domain: { }5,1, 1.4- Range: { }1 b. Domain: [ )0, ¥ Range: [ )2, ¥
Loading page 19...
ML-14
Mini Lecture 2.3
The Algebra of Functions
Learning Objectives:
1. Find the domain of a function.
2. Use the algebra of functions to combine functions and determine domains.
Examples:
State the domain of each function.
1. a. ( ) 3 1f x x= - b. ( ) 4
2
x
g x x
= - c. ( ) 2
6
h x x x
= + - d. ( ) 1 7
5 9
p x x x
= +
+ -
2. Let ( ) xxxf 22 −= and ( ) 3+= xxg . Find the following;
a. ( )( )xgf + b. the domain of gf + c. ( )( )2−+ gf
3. Let ( ) 2
5
+
= x
xf and ( ) 6
1
g x x
= - . Find the following;
a. ( )( )xgf + b. The domain of gf
Mini Lecture 2.3
The Algebra of Functions
Learning Objectives:
1. Find the domain of a function.
2. Use the algebra of functions to combine functions and determine domains.
Examples:
State the domain of each function.
1. a. ( ) 3 1f x x= - b. ( ) 4
2
x
g x x
= - c. ( ) 2
6
h x x x
= + - d. ( ) 1 7
5 9
p x x x
= +
+ -
2. Let ( ) xxxf 22 −= and ( ) 3+= xxg . Find the following;
a. ( )( )xgf + b. the domain of gf + c. ( )( )2−+ gf
3. Let ( ) 2
5
+
= x
xf and ( ) 6
1
g x x
= - . Find the following;
a. ( )( )xgf + b. The domain of gf
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ML-15
Mini Lecture 2.4
Linear Functions and Slope
Learning Objectives:
1. Use intercepts to graph a linear function in standard form.
2. Compute a line’s slope.
3. Find a line’s slope and y-intercept from its equation.
4. Graph linear functions in slope-intercept form.
5. Graph horizontal or vertical lines.
6. Interpret slope as rate of change.
7. Find a function’s average rate of change.
8. Use slope and y-intercept to model data.
Examples:
1. Use intercepts and a checkpoint to graph each linear function. Name the x-intercept and the
y-intercept.
a. 1052 =+ yx b. – 2 4x y =
2. Find the slope of the line passing through each pair of points. Then indicate whether the line
through the points rises, falls, is horizontal, or is vertical.
a. (2, 5) and (–6, 3) b. )3,1(and)0,5(
c. (3, 0) and (3, 4)- d. (2, 4) and ( 6, 4)-
3. a. Find the slope and y-intercept for the line whose equation is 1243 =+ yx and then graph
the equation.
b. Find the slope and y-intercept for the linear function 3
2
1
)( += xxf and then graph the
function.
4. Graph the linear equations.
a. 2=x b. 12–3 =y
Teaching Notes:
• The standard form of the equation of a line is CByAx =+ , as long as A and B are not
both zero.
• A x-intercept will have a corresponding y coordinate of 0.
• A y-intercept will have a corresponding x coordinate of 0.
• The slope of a line compares the vertical change to the horizontal change ( )run
rise .
• Slope formula is:
12
12
–
–
xx
yy
m = .
• A line that rises from left to right has a positive slope.
• A line that falls from left to right has a negative slope.
• A line that is horizontal has zero slope.
• A line that is vertical has an undefined slope.
• The slope-intercept form of the equation of a line is bmxy += where m is the slope and
b is the y-intercept.
Mini Lecture 2.4
Linear Functions and Slope
Learning Objectives:
1. Use intercepts to graph a linear function in standard form.
2. Compute a line’s slope.
3. Find a line’s slope and y-intercept from its equation.
4. Graph linear functions in slope-intercept form.
5. Graph horizontal or vertical lines.
6. Interpret slope as rate of change.
7. Find a function’s average rate of change.
8. Use slope and y-intercept to model data.
Examples:
1. Use intercepts and a checkpoint to graph each linear function. Name the x-intercept and the
y-intercept.
a. 1052 =+ yx b. – 2 4x y =
2. Find the slope of the line passing through each pair of points. Then indicate whether the line
through the points rises, falls, is horizontal, or is vertical.
a. (2, 5) and (–6, 3) b. )3,1(and)0,5(
c. (3, 0) and (3, 4)- d. (2, 4) and ( 6, 4)-
3. a. Find the slope and y-intercept for the line whose equation is 1243 =+ yx and then graph
the equation.
b. Find the slope and y-intercept for the linear function 3
2
1
)( += xxf and then graph the
function.
4. Graph the linear equations.
a. 2=x b. 12–3 =y
Teaching Notes:
• The standard form of the equation of a line is CByAx =+ , as long as A and B are not
both zero.
• A x-intercept will have a corresponding y coordinate of 0.
• A y-intercept will have a corresponding x coordinate of 0.
• The slope of a line compares the vertical change to the horizontal change ( )run
rise .
• Slope formula is:
12
12
–
–
xx
yy
m = .
• A line that rises from left to right has a positive slope.
• A line that falls from left to right has a negative slope.
• A line that is horizontal has zero slope.
• A line that is vertical has an undefined slope.
• The slope-intercept form of the equation of a line is bmxy += where m is the slope and
b is the y-intercept.
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ML-16
Answers: 1. a. b.
2. a. 4
1
=m , rises b. 3
4
m = - , falls c. undefined, vertical d. 0, horizontal
3. a. 3)(0,intercept-
4
3
–3
4
3
– ymxy =+=
b. 1
2
m = - y-intercept (0,3)
4. a. b.
Answers: 1. a. b.
2. a. 4
1
=m , rises b. 3
4
m = - , falls c. undefined, vertical d. 0, horizontal
3. a. 3)(0,intercept-
4
3
–3
4
3
– ymxy =+=
b. 1
2
m = - y-intercept (0,3)
4. a. b.
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ML-17
Mini Lecture 2.5
The Point-Slope Form of the Equation of a Line
Learning Objectives:
1. Use the point-slope form to write equations of a line.
2. Model data with linear functions and make predictions.
3. Find slopes and equations of parallel and perpendicular lines.
Examples:
1. Write an equation in point-slope form for the line with the given information.
a. slope = –3, passing through (–4, 1)
b. slope = 3
2 , passing through (6, 3)
c. slope = 0, passing through (–3, 2)
2. Write an equation in point slope form for the line with the given information. Then write
the equation in slope-intercept form.
a. slope = 4
1 , passing through (4, 3)
b. passing through (1, 5) and (3, –5)
c. passing through (–2, 4) and (2, 6)
3. Find the slope of a line parallel to each given line.
a. 7
4
3 += xy b. 63–2 =yx c. 82 =+ yx
4. Find the slope of a line perpendicular to each given line.
a. 14– += xy b. 123 =+ yx c. 102–4 =yx
Write an equation for each of the following in slope-intercept form.
5. A line passing through the origin (0, 0) and parallel to a line whose equation is 5–4xy = .
6. A line passing through (4, –4) and parallel to a line whose equation is 62 =+ yx .
7. A line passing through (6, –1) and perpendicular to a line whose equation is 13– =yx .
8. A line passing through (2, –5) and perpendicular to a line whose equation is 072–4 =+xy .
Teaching Notes:
• The point-slope form of the equation of a non-vertical line with slope m that passes
through the point ( )yx, is ( )11 xxmyy −=− .
• Make sure students memorize the point-slope form of a linear equation and know what
each letter represents.
• Students will need lots of practice on this.
• This is a good time for students to be able to visualize parallel line lines and “see” that
they have the same slopes, but different y-intercepts.
Mini Lecture 2.5
The Point-Slope Form of the Equation of a Line
Learning Objectives:
1. Use the point-slope form to write equations of a line.
2. Model data with linear functions and make predictions.
3. Find slopes and equations of parallel and perpendicular lines.
Examples:
1. Write an equation in point-slope form for the line with the given information.
a. slope = –3, passing through (–4, 1)
b. slope = 3
2 , passing through (6, 3)
c. slope = 0, passing through (–3, 2)
2. Write an equation in point slope form for the line with the given information. Then write
the equation in slope-intercept form.
a. slope = 4
1 , passing through (4, 3)
b. passing through (1, 5) and (3, –5)
c. passing through (–2, 4) and (2, 6)
3. Find the slope of a line parallel to each given line.
a. 7
4
3 += xy b. 63–2 =yx c. 82 =+ yx
4. Find the slope of a line perpendicular to each given line.
a. 14– += xy b. 123 =+ yx c. 102–4 =yx
Write an equation for each of the following in slope-intercept form.
5. A line passing through the origin (0, 0) and parallel to a line whose equation is 5–4xy = .
6. A line passing through (4, –4) and parallel to a line whose equation is 62 =+ yx .
7. A line passing through (6, –1) and perpendicular to a line whose equation is 13– =yx .
8. A line passing through (2, –5) and perpendicular to a line whose equation is 072–4 =+xy .
Teaching Notes:
• The point-slope form of the equation of a non-vertical line with slope m that passes
through the point ( )yx, is ( )11 xxmyy −=− .
• Make sure students memorize the point-slope form of a linear equation and know what
each letter represents.
• Students will need lots of practice on this.
• This is a good time for students to be able to visualize parallel line lines and “see” that
they have the same slopes, but different y-intercepts.
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ML-18
Answers:
1. a. )4(3–1– += xy b. 6–(
3
2
3– xy = ) c. )3(02– += xy 2. a. )4–(
4
1
3– xy = ;
2
4
1 += xy b. )1–(5–5– xy = or )3–(5–5 xy =+ ; 105– += xy c. )2–(
2
1
6– xy = or
)2(
2
1
4– += xy ; 5
2
1 += xy 3. a. 4
3 b. 3
2 c. 2
1– 4. a. 4
1 b. 3
1 c. 2
1– 5. xy 4=
6. 2–
2
1– xy = 7. 173– += xy 8. 1–2– xy =
Answers:
1. a. )4(3–1– += xy b. 6–(
3
2
3– xy = ) c. )3(02– += xy 2. a. )4–(
4
1
3– xy = ;
2
4
1 += xy b. )1–(5–5– xy = or )3–(5–5 xy =+ ; 105– += xy c. )2–(
2
1
6– xy = or
)2(
2
1
4– += xy ; 5
2
1 += xy 3. a. 4
3 b. 3
2 c. 2
1– 4. a. 4
1 b. 3
1 c. 2
1– 5. xy 4=
6. 2–
2
1– xy = 7. 173– += xy 8. 1–2– xy =
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ML-19
Mini Lecture 3.1
Systems of Linear Equations in Two Variables
Learning Objectives:
1. Determine whether an ordered pair is a solution of a system of linear equations.
2. Solve systems of linear equations by graphing.
3. Solve systems of linear equations by substitution.
4. Solve systems of linear equations by addition.
5. Select the most efficient method for solving a system of linear equations.
6. Identify systems that do not have exactly one ordered-pair solution.
Examples:
1. Determine whether (2, –3) is a solution of the system.
2 4
2 1
x y
x y
ì + =ïïí
ï + = -ïî
2. Solve by graphing: –3 0
2
x y
y x
ì + =ïïí
ï = +ïî
3. Solve by the substitution method.
a. 2 9
3 4 9
y x
x y
ì = +ïïí
ï - =ïî
b. 3 5 12
4 11
x y
x y
ì + =ïïí
ï + =ïî
4. Solve by the addition method.
a. 2 –9
3 5 4
x y
x y
ì + =ïïí
ï + =ïî
b.
5 3
4 2
1
2 6 6
x y
x y
+ =
= = -
5. Solve by the method of your choice. Identify inconsistent systems and systems with
dependent equations.
a. – 2 1
3 6 2
x y
x y
ì =ïïí
ï - =ïî
b. 3 – –3
6 2 6
x y
x y
ì =ïïí
ï- = =ïî
Teaching Notes:
• A system of linear equations in two variables represents a pair of lines. There are three
possibilities for solutions:
a. If two lines intersect at one point, then there is exactly one ordered-pair solution.
b. If two lines are parallel, then there is no solution.
c. If two lines are identical, then there are infinitely many solutions.
• All three methods for solving systems of linear equations in two variables will produce
the same answer; however, one method will sometimes be more efficient than another.
Answers: 1. no 2. (1, 3) 3. a. (–9, –9)
Mini Lecture 3.1
Systems of Linear Equations in Two Variables
Learning Objectives:
1. Determine whether an ordered pair is a solution of a system of linear equations.
2. Solve systems of linear equations by graphing.
3. Solve systems of linear equations by substitution.
4. Solve systems of linear equations by addition.
5. Select the most efficient method for solving a system of linear equations.
6. Identify systems that do not have exactly one ordered-pair solution.
Examples:
1. Determine whether (2, –3) is a solution of the system.
2 4
2 1
x y
x y
ì + =ïïí
ï + = -ïî
2. Solve by graphing: –3 0
2
x y
y x
ì + =ïïí
ï = +ïî
3. Solve by the substitution method.
a. 2 9
3 4 9
y x
x y
ì = +ïïí
ï - =ïî
b. 3 5 12
4 11
x y
x y
ì + =ïïí
ï + =ïî
4. Solve by the addition method.
a. 2 –9
3 5 4
x y
x y
ì + =ïïí
ï + =ïî
b.
5 3
4 2
1
2 6 6
x y
x y
+ =
= = -
5. Solve by the method of your choice. Identify inconsistent systems and systems with
dependent equations.
a. – 2 1
3 6 2
x y
x y
ì =ïïí
ï - =ïî
b. 3 – –3
6 2 6
x y
x y
ì =ïïí
ï- = =ïî
Teaching Notes:
• A system of linear equations in two variables represents a pair of lines. There are three
possibilities for solutions:
a. If two lines intersect at one point, then there is exactly one ordered-pair solution.
b. If two lines are parallel, then there is no solution.
c. If two lines are identical, then there are infinitely many solutions.
• All three methods for solving systems of linear equations in two variables will produce
the same answer; however, one method will sometimes be more efficient than another.
Answers: 1. no 2. (1, 3) 3. a. (–9, –9)
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ML-20
Mini Lecture 3.2
Problem Solving and Business Applications Using Systems of Equations
Learning Objectives:
1. Solve problems using systems of equations.
2. Use functions to model revenue, cost, and profit, and perform a break even analysis.
Examples:
1. At JoJo’s pet and supply store female gerbils sell for $8 each and male gerbils sell for $5
each. On Saturday 20 gerbils were sold for a total of $139. How many males and how many
females were sold on Saturday?
2. To promote the grand opening of Leroy’s Toys the store gave away 9000 miniature cars and
goofy sunglasses. The cars cost 13¢ and the sunglasses cost 15¢ each. Leroy spent a total of
$1290 on the giveaways. How many of each item did he buy?
3. The larger of two numbers is equal to three times the smaller. If twice the larger is added to
three times the smaller, the sum is 27. Find the numbers.
4. Twice the length of a rectangle is equal to five times its width. The perimeter of the rectangle
is 77 inches. Find the dimensions of the rectangle.
5. Cashews cost $3.60 per pound and almonds cost $2.70 per pound. For a fundraiser, the
volleyball team will be selling bags of mixed nuts. How many pounds of cashews and how
many pounds of almonds should the team buy in order to make a 60 pound mixture that will
sell for $3.00 per pound?
6. How many gallons of 15% alcohol solution and how many gallons of 40% alcohol solution
should be mixed to get 20 gallons of a 30% alcohol solution?
7. Joe and Jack inherited $200,000 from their Aunt Lulu. They each decided to put their money
into savings accounts for 1 year and then decide how to spend it. Joe’s money earned 5%
interest and Jack’s earned 3.8%. Together, their money earned $8,608 in interest. How much
did each boy inherit?
8. A small airplane can travel 600 miles in 4 hours with the wind. The return trip against the
wind takes 5 hours. Find the speed of the plan in still air and the speed of the wind.
9. Since Jane’s grandparents enjoy making birdhouses and selling them at local craft shows.
They will pay $150 for the booth rental for the weekend. The materials for making each
birdhouse cost $8.75. If they are able to sell the birdhouses for $15.00 each, how many will
they need to sell to beak even? What if they sell 15 birdhouses? 40 birdhouses?
Answers: 1. 13 females; 7 males 2. 6000 sunglasses; 3000 cars 3. 3 and 9 4. 11 inches wide;
27.5 inches long 5. 20 lbs. cashews’ 40 lbs. almonds 6. 12 gallons of 40%; 8 gallons of 15%
7. Joe $84,000; Jack $116,000 8. 135 mph; 15 mph wind 9. 24 birdhouses to break even; they will
lose $56.25; they will make $100.
Mini Lecture 3.2
Problem Solving and Business Applications Using Systems of Equations
Learning Objectives:
1. Solve problems using systems of equations.
2. Use functions to model revenue, cost, and profit, and perform a break even analysis.
Examples:
1. At JoJo’s pet and supply store female gerbils sell for $8 each and male gerbils sell for $5
each. On Saturday 20 gerbils were sold for a total of $139. How many males and how many
females were sold on Saturday?
2. To promote the grand opening of Leroy’s Toys the store gave away 9000 miniature cars and
goofy sunglasses. The cars cost 13¢ and the sunglasses cost 15¢ each. Leroy spent a total of
$1290 on the giveaways. How many of each item did he buy?
3. The larger of two numbers is equal to three times the smaller. If twice the larger is added to
three times the smaller, the sum is 27. Find the numbers.
4. Twice the length of a rectangle is equal to five times its width. The perimeter of the rectangle
is 77 inches. Find the dimensions of the rectangle.
5. Cashews cost $3.60 per pound and almonds cost $2.70 per pound. For a fundraiser, the
volleyball team will be selling bags of mixed nuts. How many pounds of cashews and how
many pounds of almonds should the team buy in order to make a 60 pound mixture that will
sell for $3.00 per pound?
6. How many gallons of 15% alcohol solution and how many gallons of 40% alcohol solution
should be mixed to get 20 gallons of a 30% alcohol solution?
7. Joe and Jack inherited $200,000 from their Aunt Lulu. They each decided to put their money
into savings accounts for 1 year and then decide how to spend it. Joe’s money earned 5%
interest and Jack’s earned 3.8%. Together, their money earned $8,608 in interest. How much
did each boy inherit?
8. A small airplane can travel 600 miles in 4 hours with the wind. The return trip against the
wind takes 5 hours. Find the speed of the plan in still air and the speed of the wind.
9. Since Jane’s grandparents enjoy making birdhouses and selling them at local craft shows.
They will pay $150 for the booth rental for the weekend. The materials for making each
birdhouse cost $8.75. If they are able to sell the birdhouses for $15.00 each, how many will
they need to sell to beak even? What if they sell 15 birdhouses? 40 birdhouses?
Answers: 1. 13 females; 7 males 2. 6000 sunglasses; 3000 cars 3. 3 and 9 4. 11 inches wide;
27.5 inches long 5. 20 lbs. cashews’ 40 lbs. almonds 6. 12 gallons of 40%; 8 gallons of 15%
7. Joe $84,000; Jack $116,000 8. 135 mph; 15 mph wind 9. 24 birdhouses to break even; they will
lose $56.25; they will make $100.
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ML-21
Mini Lecture 3.3
Systems of Linear Equations in Three Variables
Learning Objectives:
1. Verify the solution of a system of linear equations in three variables.
2. Solve systems of linear equations in three variables.
3. Identify inconsistent and dependent systems.
4. Solve problems using systems in three variables.
Examples
1. Show that the ordered triple (1, 2, 3) is a solution of the system:
6
2 3
2 3 4
x y z
x y z
x y z
ì + + =ïïïï - + =í
ïï + - = -ïïî
2. Solve the system:
a.
4
2
2 2 2
x y z
x y z
x y z
ì + + =ïïïï - - =í
ïï + - =ïïî
b.
– 4 – 3 2
2 8 6 1
3 0
x y z
x y z
x y z
ì + =ïïïï - + =í
ïï - + =ïïî
c.
2 3 – 5
4 6 2 10
4 3 5
x y z
x y z
x y z
ì + =ïïïï + - =í
ïï - + =ïïî
3. Create three equations from the stated problem and then solve.
The sum of the three numbers is 14. The largest is 4 times the smallest, while the sum of
the smallest and twice the largest is 18.
4. Find the quadratic function 2
y ax bx c= = = whose graph passes through the points
(1, 3), (2, 5), and ( – 1, 11).
Teaching Notes:
• A system of linear equations is three variables represents three planes.
• A linear system that intersects at one point is called a consistent system and has an
ordered triple as an answer (x, y, z).
• A linear system that intersects at infinitely many points is also called a consistent system
and is also called
Mini Lecture 3.3
Systems of Linear Equations in Three Variables
Learning Objectives:
1. Verify the solution of a system of linear equations in three variables.
2. Solve systems of linear equations in three variables.
3. Identify inconsistent and dependent systems.
4. Solve problems using systems in three variables.
Examples
1. Show that the ordered triple (1, 2, 3) is a solution of the system:
6
2 3
2 3 4
x y z
x y z
x y z
ì + + =ïïïï - + =í
ïï + - = -ïïî
2. Solve the system:
a.
4
2
2 2 2
x y z
x y z
x y z
ì + + =ïïïï - - =í
ïï + - =ïïî
b.
– 4 – 3 2
2 8 6 1
3 0
x y z
x y z
x y z
ì + =ïïïï - + =í
ïï - + =ïïî
c.
2 3 – 5
4 6 2 10
4 3 5
x y z
x y z
x y z
ì + =ïïïï + - =í
ïï - + =ïïî
3. Create three equations from the stated problem and then solve.
The sum of the three numbers is 14. The largest is 4 times the smallest, while the sum of
the smallest and twice the largest is 18.
4. Find the quadratic function 2
y ax bx c= = = whose graph passes through the points
(1, 3), (2, 5), and ( – 1, 11).
Teaching Notes:
• A system of linear equations is three variables represents three planes.
• A linear system that intersects at one point is called a consistent system and has an
ordered triple as an answer (x, y, z).
• A linear system that intersects at infinitely many points is also called a consistent system
and is also called
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ML-22
Mini Lecture 3.4
Matrix Solutions to Linear Systems
Learning Objectives:
1. Write the augmented matrix for a linear system.
2. Perform matrix row operations.
3. Use matrices to solve linear systems in two variables.
4. Use matrices to solve linear systems in three variables.
5. Use matrices to identify inconsistent and dependent systems.
Examples:
1. Write the matrix for each system.
a. 3 – 4 12
6 5
x y
x y
ì =ïïí
ï + =ïî
b. 2 7
4 8
x y
x
ì + =ïïí
ï =ïî
c.
2 3 0
3 4 3
2 5 10
x y z
x y z
x y z
ì + + =ïïïï - + =í
ïï + - =ïïî
2. Write the system of linear equations represented by each augmented matrix.
a.
11
1
12
1–1 b.
0
6
11
3–2 c.
1–
11–
3
221
2
Mini Lecture 3.4
Matrix Solutions to Linear Systems
Learning Objectives:
1. Write the augmented matrix for a linear system.
2. Perform matrix row operations.
3. Use matrices to solve linear systems in two variables.
4. Use matrices to solve linear systems in three variables.
5. Use matrices to identify inconsistent and dependent systems.
Examples:
1. Write the matrix for each system.
a. 3 – 4 12
6 5
x y
x y
ì =ïïí
ï + =ïî
b. 2 7
4 8
x y
x
ì + =ïïí
ï =ïî
c.
2 3 0
3 4 3
2 5 10
x y z
x y z
x y z
ì + + =ïïïï - + =í
ïï + - =ïïî
2. Write the system of linear equations represented by each augmented matrix.
a.
11
1
12
1–1 b.
0
6
11
3–2 c.
1–
11–
3
221
2
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ML-23
Answers: 1. a.
5
12
61
4–3 b.
8
7
04
12 c.
10
3
0
5–12
14–3
321
2. a. 112
1–
=+
=
yx
yx
b. 0
63–2
=+
=
yx
yx c.
1–22
11–2––
34
=++
=+
=++
zyx
zyx
zyx
3. a.
13
4–
3–2
21 b.
21
4–
7–0
21
c.
8–
9
2
2–31
430
121
4. a. (–3, 1) b. (1, 2, –2) c. (–2, 3, –1)
Answers: 1. a.
5
12
61
4–3 b.
8
7
04
12 c.
10
3
0
5–12
14–3
321
2. a. 112
1–
=+
=
yx
yx
b. 0
63–2
=+
=
yx
yx c.
1–22
11–2––
34
=++
=+
=++
zyx
zyx
zyx
3. a.
13
4–
3–2
21 b.
21
4–
7–0
21
c.
8–
9
2
2–31
430
121
4. a. (–3, 1) b. (1, 2, –2) c. (–2, 3, –1)
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ML-24
Mini Lecture 3.5
Determinants and Cramer’s Rule
Learning Objectives:
1. Evaluate a second-order determinant.
2. Solve a system of linear equations in two variables using Cramer’s rule.
3. Evaluate a third-order determinant.
4. Solve a system of linear equations in three variables using Cramer’s rule.
5. Use determinants to identify inconsistent and dependent systems.
Examples:
1. Evaluate the determinant of each of the following matrices:
a.
32
01 b.
52–
3–8
2. Use Cramer’s rule to solve the system:
2 – 3 3
4 2 10
x y
x y
ì =ïïí
ï - =ïî
3. Evaluate the determinant of the following matrix:
210
2–01
01–2
4. Use Cramer’s rule to solve the system:
– 3 6
2 7
2 3 4
x y z
x y z
x y z
ì + + =ïïïï + + =í
ïï + + =ïïî
5. Use Cramer’s rule to solve each system or to determine that the system is inconsistent or
contains dependent equations.
a. 2 – 4 5
2 3
x y
x y
ì =ïïí
ï- + =ïî
b. 2 6 16
3 8
x y
x y
ì + =ïïí
ï + =ïî
Teaching Notes:
• A matrix of order nm × has m rows and n columns.
• The determinant of a 2 x 2 matrix
22
11
ba
ba is denoted by
22
1
Mini Lecture 3.5
Determinants and Cramer’s Rule
Learning Objectives:
1. Evaluate a second-order determinant.
2. Solve a system of linear equations in two variables using Cramer’s rule.
3. Evaluate a third-order determinant.
4. Solve a system of linear equations in three variables using Cramer’s rule.
5. Use determinants to identify inconsistent and dependent systems.
Examples:
1. Evaluate the determinant of each of the following matrices:
a.
32
01 b.
52–
3–8
2. Use Cramer’s rule to solve the system:
2 – 3 3
4 2 10
x y
x y
ì =ïïí
ï - =ïî
3. Evaluate the determinant of the following matrix:
210
2–01
01–2
4. Use Cramer’s rule to solve the system:
– 3 6
2 7
2 3 4
x y z
x y z
x y z
ì + + =ïïïï + + =í
ïï + + =ïïî
5. Use Cramer’s rule to solve each system or to determine that the system is inconsistent or
contains dependent equations.
a. 2 – 4 5
2 3
x y
x y
ì =ïïí
ï- + =ïî
b. 2 6 16
3 8
x y
x y
ì + =ïïí
ï + =ïî
Teaching Notes:
• A matrix of order nm × has m rows and n columns.
• The determinant of a 2 x 2 matrix
22
11
ba
ba is denoted by
22
1
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ML-25
Mini Lecture 4.1
Solving Linear Inequalities
Learning Objectives:
1. Solve linear inequalities.
2. Recognize inequalities with no solution or all real numbers as solutions.
3. Solve applied problems using linear inequalities.
Examples:
1. Solve and graph the solution set on a number line. Give solutions in interval notation.
a. 2x –5 ≥3 b. 3x – 5 ≤ 6x + 4 c. 4
1+x > 4
1–2x + 8
3
d. 4(x + 1) > 4x + 2 e. 2x + 2 ≤ 2x – 2
Teaching Notes:
• One method of representing the solution set of an inequality is with interval notation.
Using the notation x ≥ –3 is expressed as [–3, ∞). The bracket indicates –3 is included in
the interval. The infinity symbol does not represent a real number and the interval
extends indefinitely to the right.
• When multiplying or dividing both sides of an inequality by a negative quantity,
remember to reverse the direction of the inequality symbol.
• When an inequality has been solved and the variable has been eliminated and the result is
a false statement, the inequality has no solution, Ø.
• When an inequality has been solved and the variable has been eliminated and the result is
a true statement, the solution for the inequality is all real numbers.
Answers:
1. a. [4, ∞) b. [–3, ∞) c. (–∞, 2
1 )
d. (–∞, ∞) e. Ø
Mini Lecture 4.1
Solving Linear Inequalities
Learning Objectives:
1. Solve linear inequalities.
2. Recognize inequalities with no solution or all real numbers as solutions.
3. Solve applied problems using linear inequalities.
Examples:
1. Solve and graph the solution set on a number line. Give solutions in interval notation.
a. 2x –5 ≥3 b. 3x – 5 ≤ 6x + 4 c. 4
1+x > 4
1–2x + 8
3
d. 4(x + 1) > 4x + 2 e. 2x + 2 ≤ 2x – 2
Teaching Notes:
• One method of representing the solution set of an inequality is with interval notation.
Using the notation x ≥ –3 is expressed as [–3, ∞). The bracket indicates –3 is included in
the interval. The infinity symbol does not represent a real number and the interval
extends indefinitely to the right.
• When multiplying or dividing both sides of an inequality by a negative quantity,
remember to reverse the direction of the inequality symbol.
• When an inequality has been solved and the variable has been eliminated and the result is
a false statement, the inequality has no solution, Ø.
• When an inequality has been solved and the variable has been eliminated and the result is
a true statement, the solution for the inequality is all real numbers.
Answers:
1. a. [4, ∞) b. [–3, ∞) c. (–∞, 2
1 )
d. (–∞, ∞) e. Ø
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