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Open intervals are crucial in various mathematical concepts because they signify a continuous range of values without including boundary points.","shortDefinition":null,"relatedTerms":[{"term":"closed interval","definition":"A closed interval is a set of real numbers that includes its endpoints, represented as [a, b], meaning both 'a' and 'b' are included in the set.","keyTermSlug":"closed-interval"},{"term":"limit point","definition":"A limit point of a set is a point where every neighborhood of that point contains at least one point from the set, signifying proximity without necessarily being included.","keyTermSlug":null},{"term":"continuity","definition":"Continuity refers to a property of functions where small changes in input result in small changes in output, often ensured when the function is defined on an open interval.","keyTermSlug":null}],"parents":[{"id":"Qa7KLJgL4vYEBuvB","type":"content"},{"id":"rCkBwFvvzA6p0LHc","type":"content"},{"id":"dHxHKW5TfQs1XnxJ","type":"content"}]},{"_id":"66beabcf813159ffbe289df1","slug":"closed-interval","subjectSlug":"introduction-to-mathematical-analysis","term":"closed interval","definition":"A closed interval is a set of real numbers that includes all numbers between two endpoints, as well as the endpoints themselves. 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This concept is essential when discussing properties of functions and continuity since closed intervals guarantee the inclusion of boundary points, which can affect the behavior of functions defined on those intervals.","shortDefinition":null,"relatedTerms":[{"term":"open interval","definition":"An open interval is a set of real numbers that does not include its endpoints, represented as (a, b).","keyTermSlug":"open-interval"},{"term":"bounded function","definition":"A bounded function is one whose values lie within a fixed range on a specified interval, which can be either closed or open.","keyTermSlug":"bounded-function"},{"term":"continuity","definition":"Continuity refers to a property of functions where small changes in input result in small changes in output, particularly significant on closed intervals.","keyTermSlug":null}],"parents":[{"id":"rCkBwFvvzA6p0LHc","type":"content"},{"id":"dHxHKW5TfQs1XnxJ","type":"content"},{"id":"zNNhLvLajLFRasFK","type":"content"}]},{"_id":"66beabf18aaa07e75cf3ff55","slug":"bounded-below","subjectSlug":"introduction-to-mathematical-analysis","term":"bounded below","definition":"A set is considered bounded below if there exists a real number that serves as a lower limit for the elements in that set. 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They help us visualize and manipulate sets of numbers on the number line. These tools are essential for solving equations and inequalities, and for representing subsets of real numbers.\n\nMastering intervals and absolute value provides a foundation for more advanced mathematical analysis. By grasping these concepts, you'll be better equipped to tackle complex problems involving real numbers and their properties in future math courses.\n\n## Interval Types and Notation\n### Types of Intervals\n\n ###### ![fiveable_image_carousel](https://fiveable.me)\n\n- Open intervals do not include their endpoints and are denoted using parentheses (a, b)\n- Closed intervals include both endpoints and are denoted using square brackets [a, b]\n- Half-open intervals include only one endpoint and are denoted using a combination of a parenthesis and a square bracket (a, b] or [a, b)\n- Intervals can be bounded having both upper and lower bounds (0, 1) or unbounded having only one or no bounds (0, ∞)\n- Intervals can be classified as finite having a specific length [0, 1] or infinite extending indefinitely in one or both directions (-∞, ∞)\n\n### Defining and Classifying Intervals\n- An interval is a set of real numbers that includes all numbers between any two numbers in the set\n- Intervals are defined by their endpoints and whether those endpoints are included or excluded\n- The type of bracket used to denote an interval indicates whether the endpoint is included (square bracket) or excluded (parenthesis)\n- Classifying intervals as open, closed, half-open, bounded, unbounded, finite, or infinite helps to understand the properties and characteristics of the set of numbers represented by the interval\n\n## Absolute Value: Concept and Geometry\n### Concept of Absolute Value\n- The absolute value of a real number is its distance from zero on the real number line, regardless of its sign\n- The absolute value of a number a is denoted as |a|\n- For any real number a, |a| ≥ 0, and |a| = 0 if and only if a = 0\n- The absolute value of a number is always non-negative, as it represents a distance (-3 and 3 both have an absolute value of 3)\n\n### Geometric Interpretation of Absolute Value\n- Geometrically, the absolute value of a number represents the length of the line segment from the origin to the point representing that number on the real number line\n- On a number line, the absolute value of a number is the distance between that number and zero, regardless of the direction\n- The graph of y = |x| is a V-shaped graph with the vertex at the origin (0, 0) and the two rays extending symmetrically in the positive y-direction\n- The graph of y = |x - a| + b represents a vertical shift of the basic absolute value graph by a units horizontally and b units vertically\n\n## Absolute Value: Solving Equations and Inequalities\n### Properties of Absolute Value\n- For any real numbers a and b, |a| = |b| if and only if a = b or a = -b\n- The absolute value of a product is equal to the product of the absolute values: |ab| = |a||b|\n- The absolute value of a quotient is equal to the quotient of the absolute values: |a/b| = |a|/|b|, where b ≠ 0\n- The absolute value of a sum is less than or equal to the sum of the absolute values: |a + b| ≤ |a| + |b| (triangle inequality)\n\n### Solving Absolute Value Equations and Inequalities\n- To solve an absolute value equation, consider the two possible cases: the positive and negative values of the expression inside the absolute value symbols\n- For example, to solve |x - 3| = 5, consider x - 3 = 5 and x - 3 = -5, which leads to x = 8 or x = -2\n- To solve an absolute value inequality, consider the distance between the expression inside the absolute value symbols and the number on the other side of the inequality sign\n- For example, to solve |x - 2| < 3, consider -3 < x - 2 < 3, which leads to -1 < x < 5\n\n## Representing Subsets of Real Numbers\n### Interval Notation\n- Interval notation is a way to represent a set of real numbers using parentheses, square brackets, and infinity symbols\n- In interval notation, a square bracket is used to indicate that the endpoint is included in the set [a, b], while a parenthesis is used to indicate that the endpoint is not included (a, b)\n- Infinity symbols (∞ and -∞) are used to represent unbounded intervals, such as (0, ∞) or (-∞, -2]\n- Examples of intervals in interval notation: [0, 1], (2, 5], (-∞, 0), [3, ∞)\n\n### Set-Builder Notation\n- Set-builder notation is a way to describe a set by stating the properties that its elements must satisfy\n- In set-builder notation, the set is denoted as {x | P(x)}, where x is a variable representing an element of the set, and P(x) is a statement describing the properties that x must satisfy to be included in the set\n- Set-builder notation can be used to represent intervals, as well as more complex subsets of real numbers that cannot be easily represented using interval notation\n- Examples of sets in set-builder notation: {x | x ∈ ℝ, x ≥ 0}, {x | x ∈ ℤ, -3 ≤ x < 5}, {x | x ∈ ℝ, x² < 9}","cheatsheet":null,"publishDate":null,"updatedAt":"2024-07-30T03:37:42.925Z","status":"PUBLISHED","images":[{"url":"https://storage.googleapis.com/static.prod.fiveable.me/search-images%2F%22Types_of_intervals_in_mathematical_analysis%3A_open_closed_half-open_bounded_unbounded_finite_infinite%22-CNX_Precalc_Figure_01_02_029n2.jpg","description":"Standard Notation for Defining Sets | College Algebra","sourceUrl":"https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/896/2016/10/18193529/CNX_Precalc_Figure_01_02_029n2.jpg","hostUrl":"https://courses.lumenlearning.com/waymakercollegealgebra/chapter/standard-notation-for-domain-and-range/","altText":null,"sectionTitle":"Types of Intervals","rank":1,"height":905,"width":975,"displayWidth":487,"displayHeight":452,"contentId":"66a86006f76386eb110b50b2","subjectId":"introduction-to-mathematical-analysis"},{"url":"https://storage.googleapis.com/static.prod.fiveable.me/search-images%2F%22Types_of_intervals_in_mathematical_analysis%3A_open_closed_half-open_bounded_unbounded_finite_infinite%22-CNX_Precalc_Figure_01_02_003.jpg","description":"Domain and Range | Algebra and Trigonometry","sourceUrl":"https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3252/2018/07/19134751/CNX_Precalc_Figure_01_02_003.jpg","hostUrl":"https://courses.lumenlearning.com/suny-osalgebratrig/chapter/domain-and-range/","altText":null,"sectionTitle":"Types of Intervals","rank":2,"height":692,"width":975,"displayWidth":487,"displayHeight":346,"contentId":"66a86006f76386eb110b50b2","subjectId":"introduction-to-mathematical-analysis"},{"url":"https://storage.googleapis.com/static.prod.fiveable.me/search-images%2F%22Types_of_intervals_in_mathematical_analysis%3A_open_closed_half-open_bounded_unbounded_finite_infinite%22-ss1.gif","description":"Properties of Real Numbers | Boundless Algebra","sourceUrl":"https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/1861/2017/06/23161730/ss1.gif","hostUrl":"https://courses.lumenlearning.com/boundless-algebra/chapter/properties-of-real-numbers/","altText":null,"sectionTitle":"Types of Intervals","rank":3,"height":185,"width":301,"displayWidth":150,"displayHeight":92,"contentId":"66a86006f76386eb110b50b2","subjectId":"introduction-to-mathematical-analysis"}],"tableOfContents":null,"meta":{"description":"Review 1.4 Intervals and Absolute Value for your test on Unit 1 – The Real Number System. 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a| + b represents a vertical shift of the basic absolute value graph by a units horizontally and b units vertically\n\n## Absolute Value: Solving Equations and Inequalities\n### Properties of Absolute Value\n- For any real numbers a and b, |a| = |b| if and only if a = b or a = -b\n- The absolute value of a product is equal to the product of the absolute values: |ab| = |a||b|\n- The absolute value of a quotient is equal to the quotient of the absolute values: |a/b| = |a|/|b|, where b ≠ 0\n- The absolute value of a sum is less than or equal to the sum of the absolute values: |a + b| ≤ |a| + |b| (triangle inequality)\n\n### Solving Absolute Value Equations and Inequalities\n- To solve an absolute value equation, consider the two possible cases: the positive and negative values of the expression inside the absolute value symbols\n- For example, to solve |x - 3| = 5, consider x - 3 = 5 and x - 3 = -5, which leads to x = 8 or x = -2\n- To solve an absolute value inequality, consider the distance between the expression inside the absolute value symbols and the number on the other side of the inequality sign\n- For example, to solve |x - 2| < 3, consider -3 < x - 2 < 3, which leads to -1 < x < 5\n\n## Representing Subsets of Real Numbers\n### Interval Notation\n- Interval notation is a way to represent a set of real numbers using parentheses, square brackets, and infinity symbols\n- In interval notation, a square bracket is used to indicate that the endpoint is included in the set [a, b], while a parenthesis is used to indicate that the endpoint is not included (a, b)\n- Infinity symbols (∞ and -∞) are used to represent unbounded intervals, such as (0, ∞) or (-∞, -2]\n- Examples of intervals in interval notation: [0, 1], (2, 5], (-∞, 0), [3, ∞)\n\n### Set-Builder Notation\n- Set-builder notation is a way to describe a set by stating the properties that its elements must satisfy\n- In set-builder notation, the set is denoted as {x | P(x)}, where x is a variable representing an element of the set, and P(x) is a statement describing the properties that x must satisfy to be included in the set\n- Set-builder notation can be used to represent intervals, as well as more complex subsets of real numbers that cannot be easily represented using interval notation\n- Examples of sets in set-builder notation: {x | x ∈ ℝ, x ≥ 0}, {x | x ∈ ℤ, -3 ≤ x < 5}, {x | x ∈ ℝ, x² < 9}","cheatsheet":null,"publishDate":null,"updatedAt":"2024-07-30T03:37:42.925Z","status":"PUBLISHED","images":[{"url":"https://storage.googleapis.com/static.prod.fiveable.me/search-images%2F%22Types_of_intervals_in_mathematical_analysis%3A_open_closed_half-open_bounded_unbounded_finite_infinite%22-CNX_Precalc_Figure_01_02_029n2.jpg","description":"Standard Notation for Defining Sets | College Algebra","sourceUrl":"https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/896/2016/10/18193529/CNX_Precalc_Figure_01_02_029n2.jpg","hostUrl":"https://courses.lumenlearning.com/waymakercollegealgebra/chapter/standard-notation-for-domain-and-range/","altText":null,"sectionTitle":"Types of Intervals","rank":1,"height":905,"width":975,"displayWidth":487,"displayHeight":452,"contentId":"66a86006f76386eb110b50b2","subjectId":"introduction-to-mathematical-analysis"},{"url":"https://storage.googleapis.com/static.prod.fiveable.me/search-images%2F%22Types_of_intervals_in_mathematical_analysis%3A_open_closed_half-open_bounded_unbounded_finite_infinite%22-CNX_Precalc_Figure_01_02_003.jpg","description":"Domain and Range | Algebra and Trigonometry","sourceUrl":"https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3252/2018/07/19134751/CNX_Precalc_Figure_01_02_003.jpg","hostUrl":"https://courses.lumenlearning.com/suny-osalgebratrig/chapter/domain-and-range/","altText":null,"sectionTitle":"Types of Intervals","rank":2,"height":692,"width":975,"displayWidth":487,"displayHeight":346,"contentId":"66a86006f76386eb110b50b2","subjectId":"introduction-to-mathematical-analysis"},{"url":"https://storage.googleapis.com/static.prod.fiveable.me/search-images%2F%22Types_of_intervals_in_mathematical_analysis%3A_open_closed_half-open_bounded_unbounded_finite_infinite%22-ss1.gif","description":"Properties of Real Numbers | Boundless Algebra","sourceUrl":"https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/1861/2017/06/23161730/ss1.gif","hostUrl":"https://courses.lumenlearning.com/boundless-algebra/chapter/properties-of-real-numbers/","altText":null,"sectionTitle":"Types of Intervals","rank":3,"height":185,"width":301,"displayWidth":150,"displayHeight":92,"contentId":"66a86006f76386eb110b50b2","subjectId":"introduction-to-mathematical-analysis"}],"tableOfContents":null,"meta":{"description":"Review 1.4 Intervals and Absolute Value for your test on Unit 1 – The Real Number System. 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This class provides important tools for tackling more advanced mathematical concepts.\n\n3. Discrete Mathematics: This course introduces mathematical reasoning, proof techniques, and set theory. It's crucial for developing the logical thinking skills needed in Analysis.\n\n## Classes similar to Introduction to Mathematical Analysis\n\n1. Real Analysis: Dives deeper into the properties of real numbers and functions. It's like Mathematical Analysis on steroids, with more advanced topics and even more rigorous proofs.\n\n2. Complex Analysis: Explores functions of complex variables. You'll apply similar concepts from Mathematical Analysis but in the fascinating world of complex numbers.\n\n3. Topology: Studies properties of spaces that are preserved under continuous deformations. It's a more abstract field that builds on the foundational concepts from Mathematical Analysis.\n\n4. Measure Theory: Delves into the theory of integration and probability. It's a natural progression from the integration concepts covered in Mathematical Analysis.\n\n5. Functional Analysis: Combines ideas from linear algebra and analysis to study vector spaces of functions. It's like the lovechild of Mathematical Analysis and Linear Algebra.\n\n## Majors related to Introduction to Mathematical Analysis\n\n1. Mathematics: Focuses on the study of quantity, structure, space, and change. Mathematical Analysis is a core component of this major, providing essential tools for advanced mathematical thinking.\n\n2. Physics: Involves understanding the fundamental principles governing the natural world. Mathematical Analysis provides the language and tools necessary for describing complex physical phenomena.\n\n3. Engineering: Applies scientific and mathematical principles to design and develop systems, structures, and processes. Mathematical Analysis is crucial for modeling and solving engineering problems.\n\n4. Economics: Studies the production, distribution, and consumption of goods and services. Mathematical Analysis is used in economic modeling and quantitative analysis of market behaviors.\n\n5. Computer Science: Deals with the theory and practice of computation and information processing. Mathematical Analysis provides the foundation for understanding algorithms, machine learning, and other advanced computing concepts.\n\n## What can you do with a degree in Introduction to Mathematical Analysis?\n\n1. Data Scientist: Analyzes complex data sets to extract meaningful insights. They use statistical techniques and machine learning algorithms to solve business problems and make predictions.\n\n2. Quantitative Analyst: Develops and implements complex mathematical models to support financial decision-making. They work in finance, using analytical skills to assess risk and pricing for various financial instruments.\n\n3. Research Mathematician: Conducts original research to advance mathematical knowledge. They work in academia or research institutions, tackling unsolved problems and developing new mathematical theories.\n\n4. Actuary: Assesses and manages financial risks for insurance companies and other organizations. They use mathematical and statistical methods to analyze the probability of future events and their financial impact.\n\n5. Operations Research Analyst: Uses advanced analytical methods to help organizations solve complex problems and make better decisions. They apply mathematical modeling techniques to optimize business processes and resource allocation.\n\n## Introduction to Mathematical Analysis FAQs\n\n1. How much time should I dedicate to studying for this course? Plan to spend at least 2-3 hours outside of class for every hour in lecture. This course requires consistent effort and practice to master the concepts.\n\n2. Are there any good online resources for extra practice? Websites like Khan Academy and MIT OpenCourseWare offer great supplementary materials. Many universities also publish their problem sets and lecture notes online.\n\n3. How important is Mathematical Analysis for graduate school in math? It's crucial if you're considering grad school in math or related fields. This course lays the groundwork for many advanced topics you'll encounter in graduate-level mathematics.\n\n4. Can I take this course if I struggled with Calculus? It might be challenging, but it's not impossible. You'll need to put in extra effort to strengthen your foundational skills, but the different approach to the material might actually help some concepts click.","emoji":"🏃🏽‍♀️‍➡️","order":null,"numResources":null,"active":true,"slug":"introduction-to-mathematical-analysis","generationMetadata":{"group":"Group 6 – add essential knowledge","level":"college undergraduate","branch":"Math","duration":"one semester","subBranch":"Math","lengthVariant":"less text","model":"opus"}},"pageParams":{"communitySlug":"introduction-to-mathematical-analysis","unitSlug":"unit-1","contentSlug":"intervals-absolute","docId":"rCkBwFvvzA6p0LHc"},"children":["$","$L1d",null,{"content":{"id":"rCkBwFvvzA6p0LHc","topics":[{"id":"eirkZk6jaMFbcCZs","name":"1.4 Intervals and Absolute Value","fullNumber":"1.4"}],"title":"1.4 Intervals and Absolute Value","desc":null,"summary":null,"type":"STUDY_GUIDE","slug":"intervals-absolute","date":null,"vimeoLiveLink":null,"url":null,"markdown":"Intervals and absolute value are key concepts in understanding the real number system. They help us visualize and manipulate sets of numbers on the number line. These tools are essential for solving equations and inequalities, and for representing subsets of real numbers.\n\nMastering intervals and absolute value provides a foundation for more advanced mathematical analysis. By grasping these concepts, you'll be better equipped to tackle complex problems involving real numbers and their properties in future math courses.\n\n## Interval Types and Notation\n### Types of Intervals\n\n ###### ![fiveable_image_carousel](https://fiveable.me)\n\n- Open intervals do not include their endpoints and are denoted using parentheses (a, b)\n- Closed intervals include both endpoints and are denoted using square brackets [a, b]\n- Half-open intervals include only one endpoint and are denoted using a combination of a parenthesis and a square bracket (a, b] or [a, b)\n- Intervals can be bounded having both upper and lower bounds (0, 1) or unbounded having only one or no bounds (0, ∞)\n- Intervals can be classified as finite having a specific length [0, 1] or infinite extending indefinitely in one or both directions (-∞, ∞)\n### Defining and Classifying Intervals\n- An interval is a set of real numbers that includes all numbers between any two numbers in the set\n- Intervals are defined by their endpoints and whether those endpoints are included or excluded\n- The type of bracket used to denote an interval indicates whether the endpoint is included (square bracket) or excluded (parenthesis)\n- Classifying intervals as open, closed, half-open, bounded, unbounded, finite, or infinite helps to understand the properties and characteristics of the set of numbers represented by the interval\n## Absolute Value: Concept and Geometry\n### Concept of Absolute Value\n- The absolute value of a real number is its distance from zero on the real number line, regardless of its sign\n- The absolute value of a number a is denoted as |a|\n- For any real number a, |a| ≥ 0, and |a| = 0 if and only if a = 0\n- The absolute value of a number is always non-negative, as it represents a distance (-3 and 3 both have an absolute value of 3)\n### Geometric Interpretation of Absolute Value\n- Geometrically, the absolute value of a number represents the length of the line segment from the origin to the point representing that number on the real number line\n- On a number line, the absolute value of a number is the distance between that number and zero, regardless of the direction\n- The graph of y = |x| is a V-shaped graph with the vertex at the origin (0, 0) and the two rays extending symmetrically in the positive y-direction\n- The graph of y = |x - a| + b represents a vertical shift of the basic absolute value graph by a units horizontally and b units vertically\n## Absolute Value: Solving Equations and Inequalities\n### Properties of Absolute Value\n- For any real numbers a and b, |a| = |b| if and only if a = b or a = -b\n- The absolute value of a product is equal to the product of the absolute values: |ab| = |a||b|\n- The absolute value of a quotient is equal to the quotient of the absolute values: |a/b| = |a|/|b|, where b ≠ 0\n- The absolute value of a sum is less than or equal to the sum of the absolute values: |a + b| ≤ |a| + |b| (triangle inequality)\n### Solving Absolute Value Equations and Inequalities\n- To solve an absolute value equation, consider the two possible cases: the positive and negative values of the expression inside the absolute value symbols\n- For example, to solve |x - 3| = 5, consider x - 3 = 5 and x - 3 = -5, which leads to x = 8 or x = -2\n- To solve an absolute value inequality, consider the distance between the expression inside the absolute value symbols and the number on the other side of the inequality sign\n- For example, to solve |x - 2| < 3, consider -3 < x - 2 < 3, which leads to -1 < x < 5\n## Representing Subsets of Real Numbers\n### Interval Notation\n- Interval notation is a way to represent a set of real numbers using parentheses, square brackets, and infinity symbols\n- In interval notation, a square bracket is used to indicate that the endpoint is included in the set [a, b], while a parenthesis is used to indicate that the endpoint is not included (a, b)\n- Infinity symbols (∞ and -∞) are used to represent unbounded intervals, such as (0, ∞) or (-∞, -2]\n- Examples of intervals in interval notation: [0, 1], (2, 5], (-∞, 0), [3, ∞)\n### Set-Builder Notation\n- Set-builder notation is a way to describe a set by stating the properties that its elements must satisfy\n- In set-builder notation, the set is denoted as {x | P(x)}, where x is a variable representing an element of the set, and P(x) is a statement describing the properties that x must satisfy to be included in the set\n- Set-builder notation can be used to represent intervals, as well as more complex subsets of real numbers that cannot be easily represented using interval notation\n- Examples of sets in set-builder notation: {x | x ∈ ℝ, x ≥ 0}, {x | x ∈ ℤ, -3 ≤ x < 5}, {x | x ∈ ℝ, x² < 9}","cheatsheet":null,"publishDate":null,"updatedAt":"2024-07-30T03:37:42.925Z","status":"PUBLISHED","images":[{"url":"https://storage.googleapis.com/static.prod.fiveable.me/search-images%2F%22Types_of_intervals_in_mathematical_analysis%3A_open_closed_half-open_bounded_unbounded_finite_infinite%22-CNX_Precalc_Figure_01_02_029n2.jpg","description":"Standard Notation for Defining Sets | College Algebra","sourceUrl":"https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/896/2016/10/18193529/CNX_Precalc_Figure_01_02_029n2.jpg","hostUrl":"https://courses.lumenlearning.com/waymakercollegealgebra/chapter/standard-notation-for-domain-and-range/","altText":null,"sectionTitle":"Types of Intervals","rank":1,"height":905,"width":975,"displayWidth":487,"displayHeight":452,"contentId":"66a86006f76386eb110b50b2","subjectId":"introduction-to-mathematical-analysis"},{"url":"https://storage.googleapis.com/static.prod.fiveable.me/search-images%2F%22Types_of_intervals_in_mathematical_analysis%3A_open_closed_half-open_bounded_unbounded_finite_infinite%22-CNX_Precalc_Figure_01_02_003.jpg","description":"Domain and Range | Algebra and Trigonometry","sourceUrl":"https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3252/2018/07/19134751/CNX_Precalc_Figure_01_02_003.jpg","hostUrl":"https://courses.lumenlearning.com/suny-osalgebratrig/chapter/domain-and-range/","altText":null,"sectionTitle":"Types of Intervals","rank":2,"height":692,"width":975,"displayWidth":487,"displayHeight":346,"contentId":"66a86006f76386eb110b50b2","subjectId":"introduction-to-mathematical-analysis"},{"url":"https://storage.googleapis.com/static.prod.fiveable.me/search-images%2F%22Types_of_intervals_in_mathematical_analysis%3A_open_closed_half-open_bounded_unbounded_finite_infinite%22-ss1.gif","description":"Properties of Real Numbers | Boundless Algebra","sourceUrl":"https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/1861/2017/06/23161730/ss1.gif","hostUrl":"https://courses.lumenlearning.com/boundless-algebra/chapter/properties-of-real-numbers/","altText":null,"sectionTitle":"Types of Intervals","rank":3,"height":185,"width":301,"displayWidth":150,"displayHeight":92,"contentId":"66a86006f76386eb110b50b2","subjectId":"introduction-to-mathematical-analysis"}],"tableOfContents":null,"meta":{"description":"Review 1.4 Intervals and Absolute Value for your test on Unit 1 – The Real Number System. 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