The document discusses operations with integers. It defines absolute value and covers addition, subtraction, multiplication, and division of integers through examples. Rules for integer operations are that the sign of the product is the product of the signs of the factors, and the sign of the quotient is the sign of the dividend. Division by zero is undefined.
Remedial instruction in education is intended to make a remedy on an existing learning issue that hinders the progress and development of learner's needed competencies and learning objectives. It tries to improve and remediate the basic skills deemed necessary amongst students.
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This is a lesson plan in Mathematics using 4 A's method.. it consist of analysis, abstraction, application and evaluation. It is good for students, that are studying math courses in line with education. This lesson plan could help them, specially those students whom slow learners because this lesson plan is in detailed format.
Remedial instruction in education is intended to make a remedy on an existing learning issue that hinders the progress and development of learner's needed competencies and learning objectives. It tries to improve and remediate the basic skills deemed necessary amongst students.
If you are looking for math video tutorials (with voice recording), you may download it on our YouTube Channel. Don't forget to SUBSCRIBE for you to get updated on our upcoming videos.
https://tinyurl.com/y9muob6q
Also, please do visit our page, LIKE and FOLLOW us on Facebook!
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This is a lesson plan in Mathematics using 4 A's method.. it consist of analysis, abstraction, application and evaluation. It is good for students, that are studying math courses in line with education. This lesson plan could help them, specially those students whom slow learners because this lesson plan is in detailed format.
sets of numbers and interval notation, operation on real numbers, simplifying expression, linear equation in one variable, aplication of linear equation in one variable, linear equation and aplication to geometry, linear inequaloties in one variable, properties of integers exponents and scientific notation
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
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The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
How to Make a Field invisible in Odoo 17Celine George
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Model Attribute Check Company Auto PropertyCeline George
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Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
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at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
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1. SECTION 1.1 Numbers
Chapter 1
Introductory Information and Review
Section 1.1:
Numbers
Types of Numbers
Order on a Number Line
Types of Numbers
Natural Numbers:
MATH 1300 Fundamentals of Mathematics
1
2. CHAPTER 1 Introductory Information and Review
Example:
Solution:
Even/Odd Natural Numbers:
2
University of Houston Department of Mathematics
3. SECTION 1.1 Numbers
Whole Numbers:
Example:
Solution:
Integers:
MATH 1300 Fundamentals of Mathematics
3
4. CHAPTER 1 Introductory Information and Review
Example:
Solution:
Even/Odd Integers:
Example:
Solution:
4
University of Houston Department of Mathematics
8. CHAPTER 1 Introductory Information and Review
Note About Division Involving Zero:
Additional Example 1:
Solution:
8
University of Houston Department of Mathematics
9. SECTION 1.1 Numbers
Additional Example 2:
Solution:
Natural Numbers:
Whole Numbers:
Integers:
Prime/Composite Numbers:
Positive/Negative Numbers:
MATH 1300 Fundamentals of Mathematics
9
10. CHAPTER 1 Introductory Information and Review
Even/Odd Numbers:
Rational Numbers:
10
University of Houston Department of Mathematics
11. SECTION 1.1 Numbers
Additional Example 3:
Solution:
Natural Numbers:
Whole Numbers:
MATH 1300 Fundamentals of Mathematics
11
12. CHAPTER 1 Introductory Information and Review
Integers:
Prime/Composite Numbers:
Positive/Negative Numbers:
Even/Odd Numbers:
Rational Numbers:
12
University of Houston Department of Mathematics
16. CHAPTER 1 Introductory Information and Review
Order on a Number Line
The Real Number Line:
Example:
Solution:
16
University of Houston Department of Mathematics
17. SECTION 1.1 Numbers
Inequality Symbols:
The following table describes additional inequality symbols.
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
17
18. CHAPTER 1 Introductory Information and Review
Example:
Solution:
Example:
Solution:
Additional Example 1:
Solution:
18
University of Houston Department of Mathematics
22. CHAPTER 1 Introductory Information and Review
22
University of Houston Department of Mathematics
23. Exercise Set 1.1: Numbers
State whether each of the following numbers is prime,
composite, or neither. If composite, then list all the
factors of the number.
1.
2.
(a) 8
(d) 7
(b) 5
(e) 12
(c) 1
(a) 11
(d) 0
(b) 6
(e) 2
(c) 15
7.
1
9
(b)
2
9
(d)
4
9
(e)
5
9
3
9
(c)
Note:
3
9
1
3
8.
Notice the pattern above and use it as a
shortcut in (f)-(m) to write the following
fractions as decimals without performing
long division.
(f)
(i)
9
9
(l)
4.
6
9
25
9
(j)
10
9
(m)
29
9
8
9
(k)
7
9
(f) 9
8
(i) 0
(k) 0.03003000300003…
14
9
Note:
6
9
9
Use the patterns from the problem above to
change each of the following decimals to either a
proper fraction or a mixed number.
(a) 0.4
(b) 0.7
(c) 2.3
(d) 1.2
(e) 4.5
(f) 7.6
Positive
Natural
Irrational
Negative
Whole
Real
Odd
Composite
Rational
Positive
Natural
Irrational
Negative
Whole
Real
Odd
Composite
Rational
Positive
Natural
Irrational
Negative
Whole
Real
Odd
Composite
Rational
Positive
Natural
Irrational
Negative
Whole
Real
2
9.
2
3
Odd
Composite
Rational
0.7
Even
Prime
Integer
Undefined
(h)
(g)
(c)
4
(e)
5
(h) 10
Even
Prime
Integer
Undefined
In (a)-(e), use long division to change the
following fractions to decimals.
(a)
(b) 0.6
(a)
Circle all of the words that can be used to describe
each of the numbers below.
Answer the following.
3.
1.3
(d)
4.7
(g) 3.1
7
(j)
9
6.
Even
Prime
Integer
Undefined
10.
4
7
Even
Prime
Integer
Undefined
Answer the following.
State whether each of the following numbers is
rational or irrational. If rational, then write the
number as a ratio of two integers. (If the number is
already written as a ratio of two integers, simply
rewrite the number.)
3
7
(a) 0.7
(b)
5
(c)
(d) 5
(e)
16
(f) 0.3
(g) 12
(h)
2.3
3.5
(j) 4
5.
(k) 0.04004000400004...
(i)
e
MATH 1300 Fundamentals of Mathematics
11. Which elements of the set
8, 2.1, 0.4, 0,
7, ,
15
4
, 5, 12 belong
to each category listed below?
(a) Even
(c) Positive
(e) Prime
(g) Natural
(i) Integer
(k) Rational
(m) Undefined
(b)
(d)
(f)
(h)
(j)
(l)
Odd
Negative
Composite
Whole
Real
Irrational
23
24. Exercise Set 1.1: Numbers
12. Which elements of the set
6.25, 4
3
4
, 3, 5, 1,
2
5
, 1, 2, 10
belong to each category listed below?
(a) Even
(c) Positive
(e) Prime
(g) Natural
(i) Integer
(k) Rational
(m) Undefined
(b)
(d)
(f)
(h)
(j)
(l)
Odd
Negative
Composite
Whole
Real
Irrational
19. Find a real number that is not a rational number.
20. Find a whole number that is not a natural
number.
21. Find a negative integer that is not a rational
number.
22. Find an integer that is not a whole number.
23. Find a prime number that is an irrational number.
24. Find a number that is both irrational and odd.
Fill in each of the following tables. Use “Y” for yes if
the row name applies to the number or “N” for no if it
does not.
Answer True or False. If False, justify your answer.j
25. All natural numbers are integers.
13.
26. No negative numbers are odd.
25
0
1
5 3
10
55
13.3
27. No irrational numbers are even.
Undefined
Natural
Whole
Integer
Rational
Irrational
Prime
Composite
Real
28. Every even number is a composite number.
29. All whole numbers are natural numbers.
30. Zero is neither even nor odd.
31. All whole numbers are integers.
14.
32. All integers are rational numbers.
2.36
0
0
5
2
2
2
7
93
Undefined
Natural
Whole
Integer
Rational
Irrational
Prime
Composite
Real
33. All nonterminating decimals are irrational
numbers.
34. Every terminating decimal is a rational number.
Answer the following.
35. List the prime numbers less than 10.
Answer the following. If no such number exists, state
“Does not exist.”
36. List the prime numbers between 20 and 30.
15. Find a number that is both prime and even.
37. List the composite numbers between 7 and 19.
16. Find a rational number that is a composite
number.
38. List the composite numbers between 31 and 41.
17. Find a rational number that is not a whole
number.
40. List the odd numbers between
39. List the even numbers between 13 and
97 .
29 and
123 .
18. Find a prime number that is negative.
24
University of Houston Department of Mathematics
25. Exercise Set 1.1: Numbers
Fill in the appropriate symbol from the set
41.
7 ______ 7
42.
3 ______
, , .
3
43. 7 ______ 7
44.
3 ______ 3
45.
81 ______ 9
46. 5 ______ 25
47. 5.32 ______
53
10
48.
7
______ 0.07
100
49.
1
3
______
1
4
50.
1
6
______
1
5
51.
1
1
______
3
4
1
1
52. ______
6
5
53.
15 ______ 4
54.
7 ______
49
55. 3 ______ 9
56.
29 ______ 5
58. Find the multiplicative inverse of the following
numbers. If undefined, write “undefined.”
(a) 3
(b) 4
(c) 1
3
(d) 2
(e) 2 7
3
59. Find the multiplicative inverse of the following
numbers. If undefined, write “undefined.”
5
(a) 2
(b) 9
(c) 0
3
(d) 1 5
(e) 1
60. Find the additive inverse of the following
numbers. If undefined, write “undefined.”
5
(a) 2
(b) 9
(c) 0
3
(d) 1 5
(e) 1
61. Place the correct number in each of the following
blanks:
(a) The sum of a number and its additive
inverse is _____. (Fill in the correct
number.)
(b) The product of a number and its
multiplicative inverse is _____. (Fill in the
correct number.)
62. Another name for the multiplicative inverse is
the ____________________.
Order the numbers in each set from least to greatest
and plot them on a number line.
(Hint: Use the approximations 2 1.41 and
3 1.73 .)
0
9
63. 1, 2, 0.4, , ,
5
4
0.49
2
64. 3 , 1 , 0.65 , , 1.5 , 0.64
3
Answer the following.
57. Find the additive inverse of the following
numbers. If undefined, write “undefined.”
(a) 3
(b) 4
(c) 1
3
2
(d) 3
(e) 2 7
MATH 1300 Fundamentals of Mathematics
25
26. CHAPTER 1 Introductory Information and Review
Section 1.2:
Integers
Operations with Integers
Operations with Integers
Absolute Value:
26
University of Houston Department of Mathematics
27. SECTION 1.2 Integers
Addition of Integers:
Example:
Solution:
Subtraction of Integers:
MATH 1300 Fundamentals of Mathematics
27
28. CHAPTER 1 Introductory Information and Review
Example:
Solution:
Multiplication of Integers:
Example:
Solution:
28
University of Houston Department of Mathematics
36. CHAPTER 1 Introductory Information and Review
Section 1.3:
Fractions
Greatest Common Divisor and Least Common Multiple
Addition and Subtraction of Fractions
Multiplication and Division of Fractions
Greatest Common Divisor and Least Common Multiple
Greatest Common Divisor:
36
University of Houston Department of Mathematics
37. SECTION 1.3 Fractions
A Method for Finding the GCD:
Least Common Multiple:
MATH 1300 Fundamentals of Mathematics
37
38. CHAPTER 1 Introductory Information and Review
A Method for Finding the LCM:
Example:
Solution:
38
University of Houston Department of Mathematics
39. SECTION 1.3 Fractions
The LCM is
Additional Example 1:
Solution:
MATH 1300 Fundamentals of Mathematics
39
40. CHAPTER 1 Introductory Information and Review
The LCM is 2 2 2 3 5 120 .
Additional Example 2:
Solution:
The LCM is 2 3 3 5 7 630 .
40
University of Houston Department of Mathematics
41. SECTION 1.3 Fractions
Additional Example 3:
Solution:
The LCM is 2 2 3 3 2 72 .
MATH 1300 Fundamentals of Mathematics
41
42. CHAPTER 1 Introductory Information and Review
Additional Example 4:
Solution:
The LCM is 2 3 3 2 5 180 .
42
University of Houston Department of Mathematics
43. SECTION 1.3 Fractions
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions with Like Denominators:
a b a b
c c
c
and
a b a b
c c
c
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
43
44. CHAPTER 1 Introductory Information and Review
Addition and Subtraction of Fractions with Unlike
Denominators:
44
University of Houston Department of Mathematics
48. CHAPTER 1 Introductory Information and Review
Solution:
(b) We must rewrite the given fractions so that they have a common denominator.
Find the LCM of the denominators 14 and 21 to find the least common denominator.
48
University of Houston Department of Mathematics
50. CHAPTER 1 Introductory Information and Review
Multiplication and Division of Fractions
Multiplication of Fractions:
50
University of Houston Department of Mathematics
58. CHAPTER 1 Introductory Information and Review
Section 1.4:
Exponents and Radicals
Evaluating Exponential Expressions
Square Roots
Evaluating Exponential Expressions
Two Rules for Exponential Expressions:
Example:
58
University of Houston Department of Mathematics
59. SECTION 1.4 Exponents and Radicals
Solution:
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
59
60. CHAPTER 1 Introductory Information and Review
Additional Properties for Exponential Expressions:
Two Definitions:
Quotient Rule for Exponential Expressions:
Exponential Expressions with Bases of Products:
Exponential Expressions with Bases of Fractions:
Example:
Evaluate each of the following:
(a) 2 3
(b)
59
56
2
(c)
5
3
Solution:
60
University of Houston Department of Mathematics
66. CHAPTER 1 Introductory Information and Review
Square Roots
Definitions:
Two Rules for Square Roots:
Writing Radical Expressions in Simplest Radical Form:
66
University of Houston Department of Mathematics
67. SECTION 1.4 Exponents and Radicals
Example:
Solution:
Example:
MATH 1300 Fundamentals of Mathematics
67
68. CHAPTER 1 Introductory Information and Review
Solution:
Exponential Form:
Additional Example 1:
Solution:
68
University of Houston Department of Mathematics
69. SECTION 1.4 Exponents and Radicals
Additional Example 2:
MATH 1300 Fundamentals of Mathematics
69
70. CHAPTER 1 Introductory Information and Review
Solution:
Additional Example 3:
Solution:
70
University of Houston Department of Mathematics
73. Exercise Set 1.4: Exponents and Radicals
Evaluate the following.
2 2
(b) 6
2
23
29. (a)
28
30. (a)
42.
5 1
52
(b)
51
53
2
(b)
32. (a)
3
(b)
5a 2b2
44.
2
6a b
2
3
2
1 2
c d 0
3a3b6
43. 3 2
2a b
31. (a)
2
3 0
c0 d 0
2
3 1
Write each of the following expressions in simplest
radical form or as a rational number (if appropriate).
If it is already in simplest radical form, say so.
35.
36.
37.
x
45
1
2
(b)
7
(c)
18
(b)
49
(c)
32
(b)
14
(c)
81
16
(b)
16
49
(c)
55
(b)
72
(c)
27
(b)
48
(c)
500
54
(b)
80
(c)
60
52. (a)
120
(b)
(c)
84
53. (a)
1
5
3 2
(b)
4
(c)
2
7
54. (a)
1
3
3x y z
3 4 2 3
6x
y z
1
2
x x
x
7
1
2
x 2 x 3 x 4
x 4 x 1
1
k 3m2
k 1 m 2
1
2
5 3 4 2
3 4 6 1
a 4 b 3
38.
28
51. (a)
(b)
19
50. (a)
y z
50
49. (a)
5 3 4 2
20
48. (a)
6x
(b)
36
47. (a)
34. (a)
3 4 2 3
45. (a)
46. (a)
3x y z
3
0
2 1
Simplify the following. No answers should contain
negative exponents.
33. (a)
2
1
2
1
2
3
4
c7
3 5 9
1
2
180
ab c
1
2
1
2a 4 b 3
39. 1 0 9
4 ab
5d 7 e0
40.
31 d 2 e4
41.
a 0 b0
a b
0
1
55. (a)
56. (a)
MATH 1300 Fundamentals of Mathematics
7
4
1
6
(b)
(b)
(b)
5
9
1
10
11
9
2 2
(c)
5
(c)
(c)
3
11
5
2
73
74. Exercise Set 1.4: Exponents and Radicals
58. (a)
7
2
(b)
x4 y5 z 7
(b)
3
8
(b)
3
8
(c) 3 8
4
81
(b)
4
81
(c) 4 81
65. (a)
35
63. (a)
64. (a)
57. (a)
6 1, 000, 000
2 9 5
a bc
(b)
6
1,000,000
(c) 6 1,000,000
Evaluate the following.
60. (a)
5
7
6
(b)
2
3
(b)
4
(c)
4
(c)
2
(b)
5
32
(c) 5 32
4 1
16
(b)
4
1
16
(c) 4
1
16
68. (a)
3 1
27
(b)
3
1
27
(c) 3
1
27
69. (a)
59. (a)
2
5
(b)
5
1
100,000
(b)
6
1
66. (a)
5
67. (a)
32
6
10
6
We can evaluate radicals other than square roots.
With square roots, we know, for example, that
49 7 , since 7 2 49 , and 49 is not a real
number. (There is no real number that when squared
1
100,000
(c) 5
70. (a)
6
1
1
100,000
(c) 6 1
gives a value of 49 , since 7 and 7 give a value
2
2
of 49, not 49 . The answer is a complex number,
which will not be addressed in this course.) In a
similar fashion, we can compute the following:
Cube Roots
3
125 5 , since 53 125 .
3
125 5 , since 5 125 .
3
Fourth Roots
4 10, 000 10 , since 104 10, 000 .
4
10, 000 is not a real number.
Fifth Roots
5
32 2 , since 25 32 .
5
32 2 , since 2 32 .
5
Sixth Roots
6 1
64
6
1 , since
2
1
2
6
64 .
1
64 is not a real number.
Evaluate the following. If the answer is not a real
number, state “Not a real number.”
61. (a)
(b)
64
(c) 64
62. (a)
74
64
25
(b)
25
(c) 25
University of Houston Department of Mathematics
75. SECTION 1.5 Order of Operations
Section 1.5:
Order of Operations
Evaluating Expressions Using the Order of Operations
Evaluating Expressions Using the Order of Operations
Rules for the Order of Operations:
1) Operations that are within parentheses and other grouping symbols are performed
first. These operations are performed in the order established in the following steps.
If grouping symbols are nested, evaluate the expression within the innermost
grouping symbol first and work outward.
2) Exponential expressions and roots are evaluated first.
3) Multiplication and division are performed next, moving left to right and performing
these operations in the order that they occur.
4) Addition and subtraction are performed last, moving left to right and performing
these operations in the order that they occur.
Upon removing all of the grouping symbols, repeat the steps 2 through 4 until the
final result is obtained.
MATH 1300 Fundamentals of Mathematics
75
76. CHAPTER 1 Introductory Information and Review
Example:
Solution:
Example:
Solution:
Additional Example 1:
76
University of Houston Department of Mathematics
77. SECTION 1.5 Order of Operations
Solution:
Additional Example 2:
Solution:
Additional Example 3:
Solution:
MATH 1300 Fundamentals of Mathematics
77
78. CHAPTER 1 Introductory Information and Review
Additional Example 4:
Solution:
Additional Example 5:
Solution:
78
University of Houston Department of Mathematics
81. Exercise Set 1.5: Order of Operations
Evaluate the following expressions for the given values
of the variables.
r
k
for P 5, r 1, and k 7 .
48.
x y
y z
for x 4, y 3, and z 8 .
49.
b b 2 8c
c2
for b 4 and c 2 .
50.
b b 2 4ac
2a
for a 1, b 3, and c 18 .
47. P
MATH 1300 Fundamentals of Mathematics
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82. CHAPTER 1 Introductory Information and Review
Section 1.6:
Solving Linear Equations
Linear Equations
Linear Equations
Rules for Solving Equations:
Linear Equations:
Example:
82
University of Houston Department of Mathematics
83. SECTION 1.6 Solving Linear Equations
Solution:
Example:
Solution:
Additional Example 1:
Solution:
MATH 1300 Fundamentals of Mathematics
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84. CHAPTER 1 Introductory Information and Review
Additional Example 2:
Solution:
Additional Example 3:
Solution:
84
University of Houston Department of Mathematics
85. Exercise Set 1.6: Solving Linear Equations
Solve the following equations algebraically.
1.
x 5 12
2.
23.
x 8 9
2
5
x 1 7
24. 3 x 7 2
4
25.
5 ( x 7)
3
4
9
2
5 x 1
3.
x 4 7
4.
x 2 8
26.
5.
6 x 30
27. 2
2x x 5
3x
3
7
6.
4 x 28
7.
6 x 10
28. x
x 7 5 x 1
8
6 12
8.
8 x 26
9.
3x 7 13
x 12 1 ( x 12) 3
6
10. 5 x 11 6
11. 2 x 3 4 x 7
12. 5 x 2 4 x 6
13. 3( x 2) 9 5( x 8) 3
14. 4( x 3) 5 2( x 4) 3
15. 3(2 5 x) 4(7 x 3)
16. 7 23 8 x 4 6(1 5 x)
17.
x
7
5
18.
x
10
3
19.
3
x9
2
20.
4
x 12
7
5
21. x 3
6
8
22. x 4
9
MATH 1300 Fundamentals of Mathematics
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86. CHAPTER 1 Introductory Information and Review
Section 1.7:
Interval Notation and Linear Inequalities
Linear Inequalities
Linear Inequalities
Rules for Solving Inequalities:
86
University of Houston Department of Mathematics
87. SECTION 1.7 Interval Notation and Linear Inequalities
Interval Notation:
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
87
88. CHAPTER 1 Introductory Information and Review
Example:
Solution:
Example:
88
University of Houston Department of Mathematics
89. SECTION 1.7 Interval Notation and Linear Inequalities
Solution:
Additional Example 1:
Solution:
MATH 1300 Fundamentals of Mathematics
89
90. CHAPTER 1 Introductory Information and Review
Additional Example 2:
Solution:
90
University of Houston Department of Mathematics
91. SECTION 1.7 Interval Notation and Linear Inequalities
Additional Example 3:
Solution:
Additional Example 4:
Solution:
MATH 1300 Fundamentals of Mathematics
91
92. CHAPTER 1 Introductory Information and Review
Additional Example 5:
Solution:
Additional Example 6:
Solution:
92
University of Houston Department of Mathematics
93. SECTION 1.7 Interval Notation and Linear Inequalities
Additional Example 7:
Solution:
MATH 1300 Fundamentals of Mathematics
93
94. Exercise Set 1.7: Interval Notation and Linear Inequalities
For each of the following inequalities:
(a) Write the inequality algebraically.
(b) Graph the inequality on the real number line.
(c) Write the inequality in interval notation.
Write each of the following inequalities in interval
notation.
23.
1.
x is greater than 5.
2.
x is less than or equal to 3.
4.
x is not equal to 2.
6.
x is less than 1.
8.
x is greater than or equal to 4 .
x is greater than 6 .
9.
x is not equal to 5 .
7.
x is greater than or equal to 7.
5.
x is less than 4.
3.
24.
25.
26.
27.
10. x is less than or equal to 2 .
28.
11. x is not equal to 8 .
12. x is not equal to 3.
13. x is not equal to 2 and x is not equal to 7.
Given the set S 2, 4, 3, 1 , use substitution to
3
14. x is not equal to 4 and x is not equal to 0.
determine which of the elements of S satisfy each of
the following inequalities.
29. 2 x 5 10
Write each of the following inequalities in interval
notation.
15. x 3
16. x 5
17. x 2
18. x 7
30. 4 x 2 14
31. 2 x 1 7
32. 3x 1 0
33. x 2 1 10
34.
1 2
x 5
19. 3 x 5
20. 7 x 2
21. x 7
22. x 9
For each of the following inequalities:
(a) Solve the inequality.
(b) Graph the solution on the real number line.
(c) Write the solution in interval notation.
35. 2 x 10
36. 3x 24
94
University of Houston Department of Mathematics
95. Exercise Set 1.7: Interval Notation and Linear Inequalities
37. 5 x 30
38. 4 x 40
39. 2 x 5 11
40. 3x 4 17
41. 8 3x 20
42. 10 x 0
43. 4 x 11 7 x 4
44. 5 9 x 3x 7
45. 10 x 7 2 x 6
46. 8 4 x 6 5 x
60. (a) 3 x 5
(b) 8 x 1
(c) 2 x 8
(d) 7 x 10
Answer the following.
61. You go on a business trip and rent a car for $75
per week plus 23 cents per mile. Your employer
will pay a maximum of $100 per week for the
rental. (Assume that the car rental company
rounds to the nearest mile when computing the
mileage cost.)
(a) Write an inequality that models this
situation.
(b) What is the maximum number of miles
that you can drive and still be
reimbursed in full?
47. 5 8 x 4 x 1
48. x 10 8 x 9
49. 3(4 5 x) 2(7 x)
50. 4(3 2 x) ( x 20)
51.
5
6
1 x 1 ( x 5)
3
2
52.
2
5
x 1 1 10 x
2
3
53. 10 3x 2 8
54. 9 2 x 3 13
55. 4 3 7 x 17
62. Joseph rents a catering hall to put on a dinner
theatre. He pays $225 to rent the space, and pays
an additional $7 per plate for each dinner served.
He then sells tickets for $15 each.
(a) Joseph wants to make a profit. Write an
inequality that models this situation.
(b) How many tickets must he sell to make
a profit?
63. A phone company has two long distance plans as
follows:
Plan 1: $4.95/month plus 5 cents/minute
Plan 2: $2.75/month plus 7 cents/minute
How many minutes would you need to talk each
month in order for Plan 1 to be more costeffective than Plan 2?
56. 19 5 4 x 3
57.
2
3
4
3 x1510 5
58.
3
4
562 x 5
3
Which of the following inequalities can never be true?
59. (a) 5 x 9
(b) 9 x 5
64. Craig’s goal in math class is to obtain a “B” for
the semester. His semester average is based on
four equally weighted tests. So far, he has
obtained scores of 84, 89, and 90. What range of
scores could he receive on the fourth exam and
still obtain a “B” for the semester? (Note: The
minimum cutoff for a “B” is 80 percent, and an
average of 90 or above will be considered an
“A”.)
(c) 3 x 7
(d) 5 x 3
MATH 1300 Fundamentals of Mathematics
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96. CHAPTER 1 Introductory Information and Review
Section 1.8:
Absolute Value and Equations
Absolute Value
Absolute Value
Equations of the Form |x| = C:
Special Cases for |x| = C:
Example:
96
University of Houston Department of Mathematics
97. SECTION 1.8 Absolute Value and Equations
Solution:
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
97
98. CHAPTER 1 Introductory Information and Review
Example:
Solution:
Example:
Solution:
98
University of Houston Department of Mathematics
99. SECTION 1.8 Absolute Value and Equations
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
99
100. CHAPTER 1 Introductory Information and Review
Additional Example 1:
Solution:
Additional Example 2:
Solution:
100
University of Houston Department of Mathematics
101. SECTION 1.8 Absolute Value and Equations
Additional Example 3:
Solution:
Additional Example 4:
Solution:
MATH 1300 Fundamentals of Mathematics
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102. CHAPTER 1 Introductory Information and Review
Additional Example 5:
Solution:
102
University of Houston Department of Mathematics
103. Exercise Set 1.8: Absolute Value and Equations
Solve the following equations.
1.
x 7
2.
x 5
3.
x 9
4.
x 10
5.
2 x 12
6.
3x 30
7.
x4 5
8.
x7 2
9.
x 45
10.
x 7 2
11.
3x 4 8
12.
5x 4 3
13.
3x 4 8
14.
5x 4 3
15.
2
3
x 7 1
16.
1
2
x
17.
4 3x 7 10
18.
22. 5 x 7 8
5x 2 8 2
5
6
23.
24.
3x 2 5 x 1
x 4 7x 6
1
3
19. 3 2 x 1 5 11
20. 2 2 9 x 6 4
21. 4
1
2
x 1 3 11
MATH 1300 Fundamentals of Mathematics
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