The document discusses how comparison statements are translated into mathematical inequalities. It provides examples of common comparison phrases and their inequality equivalents, such as "positive" meaning 0 < x, "greater than" meaning C < x, and "at least" meaning C ≤ x. It also covers compound statements like "x is more than a but no more than b" translating to a < x ≤ b. Various interval notations are introduced, such as (a, b] to represent the endpoints being included/excluded.
Two-Variable (Bivariate) RegressionIn the last unit, we covered LacieKlineeb
Two-Variable (Bivariate) Regression
In the last unit, we covered scatterplots and correlation. Social scientists use these as descriptive tools for getting an idea about how our variables of interest are related. But these tools only get us so far. Regression analysis is the next step. Regression is by far the most used tool in social science research.
Simple regression analysis can tell us several things:
1. Regression can estimate the relationship between x and y in their
original units of measurement. To see why this is so useful, consider the example of infant mortality and median family income. Let’s say that a policymaker is interested in knowing how much of a change in median family income is needed to significantly reduce the infant mortality rate. Correlation cannot answer this question, but regression can.
2. Regression can tell us how well the independent variable (x) explains the dependent variable (y). The measure is called the
R square.
Simple Two-Variable (Bivariate) Regression
Regression uses the equation of a line to estimate the relationship between x and y. You may remember back in algebra learning about the equation of a line. Some learned it as Y =s X + K or Y = mX + B. In statistics, we use a different form:
Equation 1: Y = B0 + B1X + u
Let’s define each term in the equation:
· Y is the dependent variable. It is placed on the Y (vertical) axis. In the example below, the dependent variable (Y) is the infant mortality rate.
· B0 is the Y intercept. B0 is also referred to as “the constant.” B0 is the point where the regression line crosses the Y axis. Importantly, B0 is equal to the
predicted value of Ywhen X=0. In most cases, B0 is does not get much attention for two reasons. First, the researcher is usually interested in the relationship between x and y. not the relationship between x and y at the single value of x=0. Second, often independent variables do not take on the value zero. Consider the AECF sample data. There are no states with low-birth-weight percentages equal to zero, so we would be extrapolating beyond what the data tell us.
· B1 is usually the main point of interest for researchers. It is the slope of the line relating x to y. Researchers usually refer to B1 as a slope coefficient, regression coefficient or simply a coefficient.
B1 measures the change in Y for a one-unit change in x. We represent change by the symbol ∆.
B1 =
· u is the error term. The error term is the distance between the regression line and the dots on the scatterplot. Think about it, regression estimates a single line through the cloud of data. Naturally, the line does not hit all the data points. The degree to which the line “misses” the data point is the error. u can also be thought of as
all the other factors that affect the infant mortality rate besides X. Importantly, we
assume that u is totally random given X.
The ...
8 Statistical SignificanceOK, measures of association are one .docxevonnehoggarth79783
8 Statistical Significance
OK, measures of association are one important thing to examine in data. There is another important thing to consider. If you find associations, are they a result of associations that exist in the population or are those associations simply a result of sampling error? Tests of statistical significance estimate the chances that your associations are a result of the population, and not simply sampling error.
Chi-Square Tests
Chi-Square tests are appropriate for nominal and ordinal variables. When you calculate Chi-Square, you determine the probability that your answer is a result of your sample and not the population. So, a probability of .05 (p = .05) means that the association that you found in your analysis would occur only 5 out of 100 times if there actually was no association in the population. If you had a p = .001, this means that out of 1,000 samples, you would find the association simply as a result of your sampling error 1 time. Convention suggests that probabilities of .05, 01, and .001 support a differing levels of statistical significance of your conclusions.
Let’s go back to our variables SEX and HAPPY. You actually do the same thing that you did when you calculated lambda, except you also check chi-square in the statistics box. Here are the exact steps:
· Analyze > Descriptive Statistics > Crosstabs
· Dependent variable as the Row variable (Mneumonic suggestion: remember DR, dependent belongs on the row)
· Independent variable as the column variable
· Statistics: Lambda or Gamma AND Chi-Square
· Continue > OK
Most of your output will look be the same as when we calculated lambda. There is one new box:
Look at the Asymp. Sig (2 sided). This means that out of 1,000 chances, 883 times you would get the lambda of 0 totally by accident of your sample! Is that statistically significant? Well, not for scientific research. People play the lottery with far worse odds than this, but remember, for a result to be statistically significant in social science research, the probability must be .05 or less.
Let’s think about our query about the relationship between gender and happiness. We have discovered that there is no association between the two and there is no statistical significance. What does that mean? Well, I guess that is a good thing for men and women. It is not a good finding however if you were expecting to find an association between the variables!
T Tests
T tests are used to determine statistical significance of scale (ratio/interval) variables. If you want to examine the associations of nominal/scale variables, you may be quickly overwhelmed with data. If you do discover that you would like to discover the statistical significance of scale variables, I would suggest that you use an Independent-sample t test. You can use your output for the Pearson’s r and it will tell you your level of significance.
Here is the output for our analysis of the association between AGE and SIBS:
SPSS has cal.
Chris Stuccio - Data science - Conversion Hotel 2015Webanalisten .nl
Slides of the keynote by Chris Stuccio (USA) at Conversion Hotel 2015, Texel, the Netherlands (#CH2015): "What’s this all about data science? Explain baysian statistics to me as a kid – what should I know?" http://conversionhotel.com
Two-Variable (Bivariate) RegressionIn the last unit, we covered LacieKlineeb
Two-Variable (Bivariate) Regression
In the last unit, we covered scatterplots and correlation. Social scientists use these as descriptive tools for getting an idea about how our variables of interest are related. But these tools only get us so far. Regression analysis is the next step. Regression is by far the most used tool in social science research.
Simple regression analysis can tell us several things:
1. Regression can estimate the relationship between x and y in their
original units of measurement. To see why this is so useful, consider the example of infant mortality and median family income. Let’s say that a policymaker is interested in knowing how much of a change in median family income is needed to significantly reduce the infant mortality rate. Correlation cannot answer this question, but regression can.
2. Regression can tell us how well the independent variable (x) explains the dependent variable (y). The measure is called the
R square.
Simple Two-Variable (Bivariate) Regression
Regression uses the equation of a line to estimate the relationship between x and y. You may remember back in algebra learning about the equation of a line. Some learned it as Y =s X + K or Y = mX + B. In statistics, we use a different form:
Equation 1: Y = B0 + B1X + u
Let’s define each term in the equation:
· Y is the dependent variable. It is placed on the Y (vertical) axis. In the example below, the dependent variable (Y) is the infant mortality rate.
· B0 is the Y intercept. B0 is also referred to as “the constant.” B0 is the point where the regression line crosses the Y axis. Importantly, B0 is equal to the
predicted value of Ywhen X=0. In most cases, B0 is does not get much attention for two reasons. First, the researcher is usually interested in the relationship between x and y. not the relationship between x and y at the single value of x=0. Second, often independent variables do not take on the value zero. Consider the AECF sample data. There are no states with low-birth-weight percentages equal to zero, so we would be extrapolating beyond what the data tell us.
· B1 is usually the main point of interest for researchers. It is the slope of the line relating x to y. Researchers usually refer to B1 as a slope coefficient, regression coefficient or simply a coefficient.
B1 measures the change in Y for a one-unit change in x. We represent change by the symbol ∆.
B1 =
· u is the error term. The error term is the distance between the regression line and the dots on the scatterplot. Think about it, regression estimates a single line through the cloud of data. Naturally, the line does not hit all the data points. The degree to which the line “misses” the data point is the error. u can also be thought of as
all the other factors that affect the infant mortality rate besides X. Importantly, we
assume that u is totally random given X.
The ...
8 Statistical SignificanceOK, measures of association are one .docxevonnehoggarth79783
8 Statistical Significance
OK, measures of association are one important thing to examine in data. There is another important thing to consider. If you find associations, are they a result of associations that exist in the population or are those associations simply a result of sampling error? Tests of statistical significance estimate the chances that your associations are a result of the population, and not simply sampling error.
Chi-Square Tests
Chi-Square tests are appropriate for nominal and ordinal variables. When you calculate Chi-Square, you determine the probability that your answer is a result of your sample and not the population. So, a probability of .05 (p = .05) means that the association that you found in your analysis would occur only 5 out of 100 times if there actually was no association in the population. If you had a p = .001, this means that out of 1,000 samples, you would find the association simply as a result of your sampling error 1 time. Convention suggests that probabilities of .05, 01, and .001 support a differing levels of statistical significance of your conclusions.
Let’s go back to our variables SEX and HAPPY. You actually do the same thing that you did when you calculated lambda, except you also check chi-square in the statistics box. Here are the exact steps:
· Analyze > Descriptive Statistics > Crosstabs
· Dependent variable as the Row variable (Mneumonic suggestion: remember DR, dependent belongs on the row)
· Independent variable as the column variable
· Statistics: Lambda or Gamma AND Chi-Square
· Continue > OK
Most of your output will look be the same as when we calculated lambda. There is one new box:
Look at the Asymp. Sig (2 sided). This means that out of 1,000 chances, 883 times you would get the lambda of 0 totally by accident of your sample! Is that statistically significant? Well, not for scientific research. People play the lottery with far worse odds than this, but remember, for a result to be statistically significant in social science research, the probability must be .05 or less.
Let’s think about our query about the relationship between gender and happiness. We have discovered that there is no association between the two and there is no statistical significance. What does that mean? Well, I guess that is a good thing for men and women. It is not a good finding however if you were expecting to find an association between the variables!
T Tests
T tests are used to determine statistical significance of scale (ratio/interval) variables. If you want to examine the associations of nominal/scale variables, you may be quickly overwhelmed with data. If you do discover that you would like to discover the statistical significance of scale variables, I would suggest that you use an Independent-sample t test. You can use your output for the Pearson’s r and it will tell you your level of significance.
Here is the output for our analysis of the association between AGE and SIBS:
SPSS has cal.
Chris Stuccio - Data science - Conversion Hotel 2015Webanalisten .nl
Slides of the keynote by Chris Stuccio (USA) at Conversion Hotel 2015, Texel, the Netherlands (#CH2015): "What’s this all about data science? Explain baysian statistics to me as a kid – what should I know?" http://conversionhotel.com
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
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Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
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Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
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Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
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Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
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
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
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
2. The following adjectives or comparison phrases are
translated into inequalities in mathematics:
Comparison Statements, Inequalities and Intervals
3. The following adjectives or comparison phrases are
translated into inequalities in mathematics:
“positive” vs. “negative”,
“non–positive” vs. ”non–negative”,
“more/greater than” vs. “less/smaller than”,
“no more/greater than” vs. “no less/smaller than”,
“at least” vs. ”at most”,
Comparison Statements, Inequalities and Intervals
4. The following adjectives or comparison phrases are
translated into inequalities in mathematics:
“positive” vs. “negative”,
“non–positive” vs. ”non–negative”,
“more/greater than” vs. “less/smaller than”,
“no more/greater than” vs. “no less/smaller than”,
“at least” vs. ”at most”,
“Positive” vs. “Negative”
Comparison Statements, Inequalities and Intervals
0
5. The following adjectives or comparison phrases are
translated into inequalities in mathematics:
“positive” vs. “negative”,
“non–positive” vs. ”non–negative”,
“more/greater than” vs. “less/smaller than”,
“no more/greater than” vs. “no less/smaller than”,
“at least” vs. ”at most”,
A quantity x is positive means that 0 < x,
0
“Positive” vs. “Negative”
Comparison Statements, Inequalities and Intervals
6. The following adjectives or comparison phrases are
translated into inequalities in mathematics:
“positive” vs. “negative”,
“non–positive” vs. ”non–negative”,
“more/greater than” vs. “less/smaller than”,
“no more/greater than” vs. “no less/smaller than”,
“at least” vs. ”at most”,
A quantity x is positive means that 0 < x,
0
+x is positive
“Positive” vs. “Negative”
Comparison Statements, Inequalities and Intervals
7. The following adjectives or comparison phrases are
translated into inequalities in mathematics:
“positive” vs. “negative”,
“non–positive” vs. ”non–negative”,
“more/greater than” vs. “less/smaller than”,
“no more/greater than” vs. “no less/smaller than”,
“at least” vs. ”at most”,
A quantity x is positive means that 0 < x,
and that x is negative means that x < 0.
0
+x is positive
“Positive” vs. “Negative”
Comparison Statements, Inequalities and Intervals
8. The following adjectives or comparison phrases are
translated into inequalities in mathematics:
“positive” vs. “negative”,
“non–positive” vs. ”non–negative”,
“more/greater than” vs. “less/smaller than”,
“no more/greater than” vs. “no less/smaller than”,
“at least” vs. ”at most”,
A quantity x is positive means that 0 < x,
and that x is negative means that x < 0.
0
+– x is negative x is positive
“Positive” vs. “Negative”
Comparison Statements, Inequalities and Intervals
9. The following adjectives or comparison phrases are
translated into inequalities in mathematics:
“positive” vs. “negative”,
“non–positive” vs. ”non–negative”,
“more/greater than” vs. “less/smaller than”,
“no more/greater than” vs. “no less/smaller than”,
“at least” vs. ”at most”,
A quantity x is positive means that 0 < x,
and that x is negative means that x < 0.
0
+– x is negative x is positive
“Positive” vs. “Negative”
Comparison Statements, Inequalities and Intervals
The phrase “the temperature T is positive” is “0 < T”.
11. A quantity x is non–positive means x is not positive,
or “x ≤ 0”,
“Non–Positive” vs. “Non–Negative”
Comparison Statements, Inequalities and Intervals
12. A quantity x is non–positive means x is not positive,
or “x ≤ 0”,
“Non–Positive” vs. “Non–Negative”
0
+–
x is non–positive
Comparison Statements, Inequalities and Intervals
13. A quantity x is non–positive means x is not positive,
or “x ≤ 0”, and that x is non–negative means x is not
negative, or “0 ≤ x”.
0
+–
x is non–positive
x is non–negative
“Non–Positive” vs. “Non–Negative”
Comparison Statements, Inequalities and Intervals
14. A quantity x is non–positive means x is not positive,
or “x ≤ 0”, and that x is non–negative means x is not
negative, or “0 ≤ x”.
0
+–
x is non–positive
x is non–negative
The phrase
“the account balance A is non–negative” is “0 ≤ A”.
“Non–Positive” vs. “Non–Negative”
Comparison Statements, Inequalities and Intervals
15. A quantity x is non–positive means x is not positive,
or “x ≤ 0”, and that x is non–negative means x is not
negative, or “0 ≤ x”.
0
+–
x is non–positive
x is non–negative
The phrase
“the account balance A is non–negative” is “0 ≤ A”.
“Non–Positive” vs. “Non–Negative”
“More/greater than” vs “Less/smaller than”
Comparison Statements, Inequalities and Intervals
16. A quantity x is non–positive means x is not positive,
or “x ≤ 0”, and that x is non–negative means x is not
negative, or “0 ≤ x”.
0
+–
x is non–positive
x is non–negative
The phrase
“the account balance A is non–negative” is “0 ≤ A”.
“Non–Positive” vs. “Non–Negative”
“More/greater than” vs “Less/smaller than”
Let C be a number, x is greater than C means
“C < x”,
C
x is more than C
Comparison Statements, Inequalities and Intervals
17. A quantity x is non–positive means x is not positive,
or “x ≤ 0”, and that x is non–negative means x is not
negative, or “0 ≤ x”.
0
+–
x is non–positive
x is non–negative
The phrase
“the account balance A is non–negative” is “0 ≤ A”.
“Non–Positive” vs. “Non–Negative”
“More/greater than” vs “Less/smaller than”
Let C be a number, x is greater than C means
“C < x”, and that x is less than C means “x < C”.
Cx is less than C
x is more than C
Comparison Statements, Inequalities and Intervals
18. “No more/greater than” vs “No less/smaller than”
and “At most” vs “At least”
Comparison Statements, Inequalities and Intervals
19. “No more/greater than” vs “No less/smaller than”
A quantity x is “no more/greater than C”
is the same as “x is at most C” and means “x ≤ C”.
and “At most” vs “At least”
Comparison Statements, Inequalities and Intervals
20. “No more/greater than” vs “No less/smaller than”
A quantity x is “no more/greater than C”
is the same as “x is at most C” and means “x ≤ C”.
+–
x is no more than C C
and “At most” vs “At least”
x is at most C
Comparison Statements, Inequalities and Intervals
21. “No more/greater than” vs “No less/smaller than”
A quantity x is “no more/greater than C”
is the same as “x is at most C” and means “x ≤ C”.
A quantity x is “no–less than C” is the same as
“x is at least C” and means “C ≤ x”.
+–
x is no more than C x is no less than CC
and “At most” vs “At least”
x is at most C x is at least C
Comparison Statements, Inequalities and Intervals
22. “No more/greater than” vs “No less/smaller than”
A quantity x is “no more/greater than C”
is the same as “x is at most C” and means “x ≤ C”.
A quantity x is “no–less than C” is the same as
“x is at least C” and means “C ≤ x”.
+–
x is no more than C x is no less than C
“The account balance A is no–less than 500”
is the same as “A is at least 500” or that “500 ≤ A”.
C
and “At most” vs “At least”
x is at most C x is at least C
Comparison Statements, Inequalities and Intervals
23. “No more/greater than” vs “No less/smaller than”
A quantity x is “no more/greater than C”
is the same as “x is at most C” and means “x ≤ C”.
A quantity x is “no–less than C” is the same as
“x is at least C” and means “C ≤ x”.
+–
x is no more than C x is no less than C
“The temperature T is no–more than 250o”
is the same as “T is at most 250o” or that “T ≤ 250o”.
“The account balance A is no–less than 500”
is the same as “A is at least 500” or that “500 ≤ A”.
C
and “At most” vs “At least”
x is at most C x is at least C
Comparison Statements, Inequalities and Intervals
24. We also have the compound statements such as
“x is more than a, but no more than b”.
Comparison Statements, Inequalities and Intervals
25. We also have the compound statements such as
“x is more than a, but no more than b”.
In inequality notation, this is “a < x ≤ b”.
Comparison Statements, Inequalities and Intervals
26. We also have the compound statements such as
“x is more than a, but no more than b”.
In inequality notation, this is “a < x ≤ b”.
+–
a a < x ≤ b b
Comparison Statements, Inequalities and Intervals
27. We also have the compound statements such as
“x is more than a, but no more than b”.
In inequality notation, this is “a < x ≤ b”.
+–
a a < x ≤ b b
and it’s denoted as: (a, b]
where “(”, “)” means the end points are excluded
and that “[”, “]” means the end points are included.
Comparison Statements, Inequalities and Intervals
28. We also have the compound statements such as
“x is more than a, but no more than b”.
In inequality notation, this is “a < x ≤ b”.
+–
a a < x ≤ b b
and it’s denoted as: (a, b]
where “(”, “)” means the end points are excluded
and that “[”, “]” means the end points are included.
A line segment as such is called an interval.
Comparison Statements, Inequalities and Intervals
29. Therefore the statement “the length L of
the stick must be more than 5 feet but no
more than 7 feet” is “5 < L ≤ 7”
We also have the compound statements such as
“x is more than a, but no more than b”.
In inequality notation, this is “a < x ≤ b”.
+–
a a < x ≤ b b
and it’s denoted as: (a, b]
where “(”, “)” means the end points are excluded
and that “[”, “]” means the end points are included.
A line segment as such is called an interval.
Comparison Statements, Inequalities and Intervals
30. Therefore the statement “the length L of
the stick must be more than 5 feet but no
more than 7 feet” is “5 < L ≤ 7”
or that L must be in the interval (5, 7].
We also have the compound statements such as
“x is more than a, but no more than b”.
In inequality notation, this is “a < x ≤ b”.
+–
a a < x ≤ b b
and it’s denoted as: (a, b]
where “(”, “)” means the end points are excluded
and that “[”, “]” means the end points are included.
A line segment as such is called an interval.
75
5 < L ≤ 7
or (5, 7]
L
Comparison Statements, Inequalities and Intervals
31. Therefore the statement “the length L of
the stick must be more than 5 feet but no
more than 7 feet” is “5 < L ≤ 7”
or that L must be in the interval (5, 7].
We also have the compound statements such as
“x is more than a, but no more than b”.
In inequality notation, this is “a < x ≤ b”.
+–
a a < x ≤ b b
and it’s denoted as: (a, b]
where “(”, “)” means the end points are excluded
and that “[”, “]” means the end points are included.
A line segment as such is called an interval.
75
5 < L ≤ 7
or (5, 7]Following is a list of interval notation.
L
Comparison Statements, Inequalities and Intervals
32. Let a, b be two numbers such that a < b, we write
ba
Comparison Statements, Inequalities and Intervals
33. Let a, b be two numbers such that a < b, we write
or a ≤ x ≤ b as [a, b],ba
Comparison Statements, Inequalities and Intervals
34. Let a, b be two numbers such that a < b, we write
or a ≤ x ≤ b as [a, b],ba
or a < x < b as (a, b),ba
Comparison Statements, Inequalities and Intervals
35. Let a, b be two numbers such that a < b, we write
or a ≤ x ≤ b as [a, b],ba
or a < x < b as (a, b),ba
Comparison Statements, Inequalities and Intervals
Note: The notation “(2, 3)”
is to be viewed as an interval or as
a point (x, y) depends on the context.
36. Let a, b be two numbers such that a < b, we write
or a ≤ x ≤ b as [a, b],ba
or a < x < b as (a, b),ba
or a ≤ x < b as [a, b),ba
or a < x ≤ b as (a, b],ba
Comparison Statements, Inequalities and Intervals
37. Using the “∞” symbol which means to “surpass all
finite numbers”, we may write the rays
∞
a
or a ≤ x, as [a, ∞),
Let a, b be two numbers such that a < b, we write
or a ≤ x ≤ b as [a, b],ba
or a < x < b as (a, b),ba
or a ≤ x < b as [a, b),ba
or a < x ≤ b as (a, b],ba
Comparison Statements, Inequalities and Intervals
38. Using the “∞” symbol which means to “surpass all
finite numbers”, we may write the rays
∞
a
or a ≤ x, as [a, ∞),
∞a
or a < x, as (a, ∞),
Let a, b be two numbers such that a < b, we write
or a ≤ x ≤ b as [a, b],ba
or a < x < b as (a, b),ba
or a ≤ x < b as [a, b),ba
or a < x ≤ b as (a, b],ba
Comparison Statements, Inequalities and Intervals
39. Using the “∞” symbol which means to “surpass all
finite numbers”, we may write the rays
∞
a
or a ≤ x, as [a, ∞),
–∞ a
or x ≤ a, as (–∞, a],
∞a
or a < x, as (a, ∞),
–∞ a
or x < a, as (–∞, a),
Let a, b be two numbers such that a < b, we write
or a ≤ x ≤ b as [a, b],ba
or a < x < b as (a, b),ba
or a ≤ x < b as [a, b),ba
or a < x ≤ b as (a, b],ba
Comparison Statements, Inequalities and Intervals
40. Intersection and Union (∩ & U) of Intervals
Comparison Statements, Inequalities and Intervals
41. Let I = [1, 3] as shown,
31
Intersection and Union (∩ & U) of Intervals
I:
and let J = (2, 4) be another interval as shown,
42
J:
Comparison Statements, Inequalities and Intervals
42. Let I = [1, 3] as shown,
31
Intersection and Union (∩ & U) of Intervals
I:
and let J = (2, 4) be another interval as shown,
42
J:
The common portion of the two intervals I and J
shown here 31
I:
42
J:
Comparison Statements, Inequalities and Intervals
43. Let I = [1, 3] as shown,
31
Intersection and Union (∩ & U) of Intervals
I:
and let J = (2, 4) be another interval as shown,
42
J:
The common portion of the two intervals I and J
shown here 31
I:
42
J:
2
3
32
Comparison Statements, Inequalities and Intervals
44. Let I = [1, 3] as shown,
31
Intersection and Union (∩ & U) of Intervals
I:
and let J = (2, 4) be another interval as shown,
42
J:
The common portion of the two intervals I and J
shown here 31
I:
42
J:
2
3
3
I ∩ J:
is called the intersection of I and J.
2
Comparison Statements, Inequalities and Intervals
45. Let I = [1, 3] as shown,
31
Intersection and Union (∩ & U) of Intervals
I:
and let J = (2, 4) be another interval as shown,
42
J:
The common portion of the two intervals I and J
shown here 31
I:
42
J:
2
3
3
I ∩ J:
is called the intersection of I and J.
It’s denoted as I ∩ J and this case I ∩ J = (2, 3].
2
Comparison Statements, Inequalities and Intervals
46. The merge of the two intervals I and J shown here
31
I:
42
J:
Comparison Statements, Inequalities and Intervals
47. The merge of the two intervals I and J shown here
31
I:
42
J:
2
3
2 41 3
Comparison Statements, Inequalities and Intervals
48. The merge of the two intervals I and J shown here
31
I:
42
J:
2
3
I U J:
is called the union of I and J and it’s denoted as I U J.
2 41 3
Comparison Statements, Inequalities and Intervals
49. The merge of the two intervals I and J shown here
31
I:
42
J:
2
3
I U J:
is called the union of I and J and it’s denoted as I U J.
2 41 3
In this case I U J = [1, 4).
Comparison Statements, Inequalities and Intervals
50. The merge of the two intervals I and J shown here
31
I:
42
J:
2
3
I U J:
is called the union of I and J and it’s denoted as I U J.
2 41 3
In this case I U J = [1, 4).
Example A. Given intervals I, J, and K, perform the
following set operation. Draw the solution and write
the answer in the interval notation.
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
a. K U J
Comparison Statements, Inequalities and Intervals
51. The merge of the two intervals I and J shown here
31
I:
42
J:
2
3
I U J:
is called the union of I and J and it’s denoted as I U J.
2 41 3
In this case I U J = [1, 4).
Example A. Given intervals I, J, and K, perform the
following set operation. Draw the solution and write
the answer in the interval notation.
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
a. K U J
–3 1
0
KWe have
Comparison Statements, Inequalities and Intervals
52. The merge of the two intervals I and J shown here
31
I:
42
J:
2
3
I U J:
is called the union of I and J and it’s denoted as I U J.
2 41 3
In this case I U J = [1, 4).
Example A. Given intervals I, J, and K, perform the
following set operation. Draw the solution and write
the answer in the interval notation.
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
a. K U J
–2
–3 1
0
K
J
We have
Comparison Statements, Inequalities and Intervals
53. The merge of the two intervals I and J shown here
31
I:
42
J:
2
3
I U J:
is called the union of I and J and it’s denoted as I U J.
2 41 3
In this case I U J = [1, 4).
Example A. Given intervals I, J, and K, perform the
following set operation. Draw the solution and write
the answer in the interval notation.
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
a. K U J
–2
–3 1
0
K
J
and K U J is
–3
0
We have
Comparison Statements, Inequalities and Intervals
the union:
54. The merge of the two intervals I and J shown here
31
I:
42
J:
2
3
I U J:
is called the union of I and J and it’s denoted as I U J.
2 41 3
In this case I U J = [1, 4).
Example A. Given intervals I, J, and K, perform the
following set operation. Draw the solution and write
the answer in the interval notation.
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
a. K U J
–2
–3 1
0
so K U J = (–3, ∞).
K
J
and K U J is
–3
0
We have
Comparison Statements, Inequalities and Intervals
the union:
55. b. K ∩ I
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
Comparison Statements, Inequalities and Intervals
56. We have 10
b. K ∩ I
–1–4 –3 K
I
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
Comparison Statements, Inequalities and Intervals
57. We have 10
b. K ∩ I
–1–4 –3
The intersection is the overlapping portion as shown
K
I
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
Comparison Statements, Inequalities and Intervals
58. We have 10
b. K ∩ I
–1–4 –3
The intersection is the overlapping portion as shown
so K ∩ I is or (–3, –1).
K
I
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
0
–1–3
Comparison Statements, Inequalities and Intervals
59. We have 10
b. K ∩ I
–1–4 –3
The intersection is the overlapping portion as shown
so K ∩ I is or (–3, –1).
K
I
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
0
–1–3
Comparison Statements, Inequalities and Intervals
Example B. Abe and Bob work at the same shop.
Abe works after 2 pm till no more than 5 pm,
Bob works from exactly 4 pm till before 7 pm,
a. draw each person's schedule on a time line and
write them using the interval notation.
60. We have 10
b. K ∩ I
–1–4 –3
The intersection is the overlapping portion as shown
so K ∩ I is or (–3, –1).
K
I
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
0
–1–3
Comparison Statements, Inequalities and Intervals
Example B. Abe and Bob work at the same shop.
Abe works after 2 pm till no more than 5 pm,
Bob works from exactly 4 pm till before 7 pm,
a. draw each person's schedule on a time line and
write them using the interval notation.
31 2 4 5 6 7 8
pm
31 2 4 5 6 7 8
pm
Abe schedule: Bob’s schedule:
61. We have 10
b. K ∩ I
–1–4 –3
The intersection is the overlapping portion as shown
so K ∩ I is or (–3, –1).
K
I
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
0
–1–3
Comparison Statements, Inequalities and Intervals
Example B. Abe and Bob work at the same shop.
Abe works after 2 pm till no more than 5 pm,
Bob works from exactly 4 pm till before 7 pm,
a. draw each person's schedule on a time line and
write them using the interval notation.
31 2 4 5 6 7 8
pm
31 2 4 5 6 7 8
pm
Abe schedule: Bob’s schedule:
62. We have 10
b. K ∩ I
–1–4 –3
The intersection is the overlapping portion as shown
so K ∩ I is or (–3, –1).
K
I
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
0
–1–3
Comparison Statements, Inequalities and Intervals
Example B. Abe and Bob work at the same shop.
Abe works after 2 pm till no more than 5 pm,
Bob works from exactly 4 pm till before 7 pm,
a. draw each person's schedule on a time line and
write them using the interval notation.
31 2 4 5 6 7 8
pm
31 2 4 5 6 7 8
pm
Abe schedule: Bob’s schedule:
A: (2, 5 ]
63. We have 10
b. K ∩ I
–1–4 –3
The intersection is the overlapping portion as shown
so K ∩ I is or (–3, –1).
K
I
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
0
–1–3
Comparison Statements, Inequalities and Intervals
Example B. Abe and Bob work at the same shop.
Abe works after 2 pm till no more than 5 pm,
Bob works from exactly 4 pm till before 7 pm,
a. draw each person's schedule on a time line and
write them using the interval notation.
31 2 4 5 6 7 8
pm
31 2 4 5 6 7 8
pm
Abe schedule: Bob’s schedule:
A: (2, 5 ]
64. We have 10
b. K ∩ I
–1–4 –3
The intersection is the overlapping portion as shown
so K ∩ I is or (–3, –1).
K
I
0
–1–4
J: x > –2 K: –3 < x ≤ 1I:
0
–1–3
Comparison Statements, Inequalities and Intervals
Example B. Abe and Bob work at the same shop.
Abe works after 2 pm till no more than 5 pm,
Bob works from exactly 4 pm till before 7 pm,
a. draw each person's schedule on a time line and
write them using the interval notation.
31 2 4 5 6 7 8
pm
31 2 4 5 6 7 8
pm
Abe schedule: Bob’s schedule:
A: (2, 5 ] B: [4, 7)
65. b. When will there someone working at the shop?
Comparison Statements, Inequalities and Intervals
66. b. When will there someone working at the shop?
Stack the schedules as shown.
Comparison Statements, Inequalities and Intervals
31 2 4 5 6 7 8 pm
31 2 4 5 6 7 8 pm
A: (2, 5 ]
B: [4, 7)
67. b. When will there someone working at the shop?
Stack the schedules as shown.
The answer is the
union of A and B
Comparison Statements, Inequalities and Intervals
31 2 4 5 6 7 8 pm
31 2 4 5 6 7 8 pm
A: (2, 5 ]
B: [4, 7)
31 2 4 5 6 7 8 pm
i.e. A U B
A U B:
68. b. When will there someone working at the shop?
Stack the schedules as shown.
The answer is the
union of A and B
Comparison Statements, Inequalities and Intervals
31 2 4 5 6 7 8 pm
31 2 4 5 6 7 8 pm
A: (2, 5 ]
B: [4, 7)
31 2 4 5 6 7 8 pm
i.e. A U B = (2, 7).
A U B:
69. b. When will there someone working at the shop?
Stack the schedules as shown.
The answer is the
union of A and B
Comparison Statements, Inequalities and Intervals
31 2 4 5 6 7 8 pm
31 2 4 5 6 7 8 pm
A: (2, 5 ]
B: [4, 7)
So there will be
someone after 2 pm till before 7 pm.
31 2 4 5 6 7 8 pm
i.e. A U B = (2, 7).
A U B:
70. b. When will there someone working at the shop?
Stack the schedules as shown.
The answer is the
union of A and B
Comparison Statements, Inequalities and Intervals
31 2 4 5 6 7 8 pm
31 2 4 5 6 7 8 pm
A: (2, 5 ]
B: [4, 7)
So there will be
someone after 2 pm till before 7 pm.
31 2 4 5 6 7 8 pm
i.e. A U B = (2, 7).
A U B:
b. When will both be working at the shop?
31 2 4 5 6 7 8 pm
31 2 4 5 6 7 8 pm
A: (2, 5 ]
B: [4, 7)
71. b. When will there someone working at the shop?
Stack the schedules as shown.
The answer is the
union of A and B
Comparison Statements, Inequalities and Intervals
31 2 4 5 6 7 8 pm
31 2 4 5 6 7 8 pm
A: (2, 5 ]
B: [4, 7)
So there will be
someone after 2 pm till before 7 pm.
31 2 4 5 6 7 8 pm
i.e. A U B = (2, 7).
A U B:
b. When will both be working at the shop?
31 2 4 5 6 7 8 pm
31 2 4 5 6 7 8 pm
A: (2, 5 ]
B: [4, 7)
31 2 4 5 6 7 8 pm
A ∩ B:
The time both
persons be working
is the intersection of
their schedule,
i.e. A ∩ B
72. b. When will there someone working at the shop?
Stack the schedules as shown.
The answer is the
union of A and B
Comparison Statements, Inequalities and Intervals
31 2 4 5 6 7 8 pm
31 2 4 5 6 7 8 pm
A: (2, 5 ]
B: [4, 7)
So there will be
someone after 2 pm till before 7 pm.
31 2 4 5 6 7 8 pm
i.e. A U B = (2, 7).
A U B:
b. When will both be working at the shop?
31 2 4 5 6 7 8 pm
31 2 4 5 6 7 8 pm
A: (2, 5 ]
B: [4, 7)
31 2 4 5 6 7 8 pm
A ∩ B:
The time both
persons be working
is the intersection of
their schedule,
i.e. A ∩ B = [4, 5]. So both be there from 4 pm to 5 pm.
73. The interval [a, a] consists of one point {x = a}.
The empty set which contains nothing is denoted as
Φ = { } and interval (a, a) = (a, a] = [a, a) = Φ.
Comparison Statements, Inequalities and Intervals
74. Exercise. A. Draw the following Inequalities. Translate each
inequality into an English phrase. (There might be more than
one way to do it)
1. x < 3 2. –5 ≤ x 3. x < –8 4. x ≤ 12
5. x ≥ 3 6. –5 > x 7. x ≥ –8 8. x > 12
Exercise. B. Translate each English phrase into an inequality.
Draw the Inequalities.
Let P be the number of people on a bus.
9. There were at least 50 people on the bus.
10. There were no more than 50 people on the bus.
11. There were less than 30 people on the bus.
12. There were no less than 28 people on the bus.
Let T be temperature outside.
13. The temperature is no more than –2o.
14. The temperature is at least than 35o.
15. The temperature is positive.
Inequalities
75. Inequalities
Let M be the amount of money I have.
16. I have at most $25.
17. I have a non–positive amount of money.
18. I have less than $45.
19. I have at least $250.
Let the basement floor number be given as a negative number
and let F be the floor number that we are on.
20. We are below the 7th floor.
21. We are above the first floor.
22. We are not below the 3rd floor basement.
24. We are on at least the 45th floor.
25. We are between the 4th floor basement and the 10th
floor.
26. We are in the basement.