1.4 Intervals and Absolute Value
4 min read•july 30, 2024
Intervals and 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.
Mastering 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.
Interval Types and Notation
Types of Intervals
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- Open intervals do not include their endpoints and are denoted using parentheses
- Closed intervals include both endpoints and are denoted using square brackets
- 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)
- Intervals can be bounded having both upper and lower bounds (0, 1) or unbounded having only one or no bounds (0, ∞)
- Intervals can be classified as finite having a specific length [0, 1] or infinite extending indefinitely in one or both directions (-∞, ∞)
Defining and Classifying Intervals
- An interval is a set of real numbers that includes all numbers between any two numbers in the set
- Intervals are defined by their endpoints and whether those endpoints are included or excluded
- The type of bracket used to denote an interval indicates whether the endpoint is included (square bracket) or excluded (parenthesis)
- 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
Absolute Value: Concept and Geometry
Concept of Absolute Value
- The absolute value of a real number is its distance from zero on the real number line, regardless of its sign
- The absolute value of a number a is denoted as |a|
- For any real number a, |a| ≥ 0, and |a| = 0 if and only if a = 0
- 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)
Geometric Interpretation of Absolute Value
- 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
- On a number line, the absolute value of a number is the distance between that number and zero, regardless of the direction
- 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
- 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
Absolute Value: Solving Equations and Inequalities
Properties of Absolute Value
- For any real numbers a and b, |a| = |b| if and only if a = b or a = -b
- The absolute value of a product is equal to the product of the absolute values: |ab| = |a||b|
- The absolute value of a quotient is equal to the quotient of the absolute values: |a/b| = |a|/|b|, where b ≠ 0
- The absolute value of a sum is less than or equal to the sum of the absolute values: |a + b| ≤ |a| + |b| ()
Solving Absolute Value Equations and Inequalities
- To solve an absolute value equation, consider the two possible cases: the positive and negative values of the expression inside the absolute value symbols
- For example, to solve , consider x - 3 = 5 and x - 3 = -5, which leads to x = 8 or x = -2
- 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
- For example, to solve , consider -3 < x - 2 < 3, which leads to -1 < x < 5
Representing Subsets of Real Numbers
Interval Notation
- Interval notation is a way to represent a set of real numbers using parentheses, square brackets, and infinity symbols
- 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)
- Infinity symbols (∞ and -∞) are used to represent unbounded intervals, such as (0, ∞) or (-∞, -2]
- Examples of intervals in interval notation: [0, 1], (2, 5], (-∞, 0), [3, ∞)
Set-Builder Notation
- Set-builder notation is a way to describe a set by stating the properties that its elements must satisfy
- 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
- 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
- Examples of sets in set-builder notation: {x | x ∈ ℝ, x ≥ 0}, {x | x ∈ ℤ, -3 ≤ x < 5}, {x | x ∈ ℝ, x² < 9}
Key Terms to Review (19)
(a, b): (a, b) represents an open interval in the real number system, indicating all real numbers x such that a < x < b. This notation captures the idea of a range of values that does not include the endpoints a and b, which are referred to as boundary points. Understanding open intervals is crucial for analyzing functions, continuity, and limits in mathematical analysis.
(a, b]: '(a, b]' is a mathematical notation that represents a specific type of interval on the real number line, including all numbers greater than 'a' and up to and including 'b'. This notation emphasizes that 'a' is not part of the interval (open at 'a'), while 'b' is included (closed at 'b'). Understanding this notation is essential when discussing intervals, boundaries, and how they relate to sets of numbers in mathematical analysis.
[a, b]: [a, b] denotes a closed interval in the context of real numbers, including all numbers x such that a ≤ x ≤ b. This means that both endpoints, a and b, are included in the interval. Closed intervals are crucial in mathematical analysis because they allow for the consideration of continuous functions over defined ranges and provide a means to discuss properties such as limits, continuity, and boundedness within those bounds.
|a| = -a if a < 0: The expression |a| = -a if a < 0 defines the absolute value of a number when that number is negative. It states that the absolute value, denoted by |a|, is equal to the negative of that number when it is less than zero, ensuring that absolute values are always non-negative. This concept connects to the broader understanding of intervals and helps to clarify how numbers are represented on the number line.
|a| = a if a ≥ 0: |a| = a if a ≥ 0 defines the absolute value of a number, which is a measure of its distance from zero on the number line, regardless of direction. This equation states that when the number 'a' is non-negative (greater than or equal to zero), its absolute value is simply the number itself. This concept helps in understanding intervals and the behavior of numbers in mathematical analysis, particularly when dealing with inequalities and distance measurements.
|x - 2| < 3: The expression |x - 2| < 3 represents an inequality involving absolute value, indicating that the distance between the variable x and the number 2 is less than 3. This means that x can take values that are within 3 units of 2 on the number line, creating an interval that captures all such numbers. Understanding this concept is crucial for working with intervals and inequalities, as it helps in visualizing how absolute value defines distances and ranges on the real number line.
|x - 3| = 5: The equation |x - 3| = 5 expresses the absolute value of the difference between a variable x and the number 3, indicating that this difference can equal either 5 or -5. Absolute value measures the distance of a number from zero on a number line, meaning x could be either 8 or -2, as both yield a distance of 5 from 3. Understanding this equation helps to solve for potential values of x by recognizing how absolute values define ranges of solutions.
Absolute Value: Absolute value is a mathematical concept that represents the distance of a number from zero on the number line, regardless of direction. It is always non-negative, meaning it cannot be less than zero, and is denoted by two vertical bars surrounding the number, like this: $$|x|$$. This concept is crucial for understanding intervals, inequalities, and functions that involve distances or magnitudes.
Bounded above: A set of numbers is said to be bounded above if there exists a real number that is greater than or equal to every number in the set. This means that all elements of the set do not exceed a certain limit, making it essential in understanding the behavior of sequences and functions. Boundedness leads to the concepts of supremum and infimum, which help in identifying the least upper bound of a set and understanding the properties of real numbers, intervals, and their absolute values.
Bounded below: A set is considered bounded below if there exists a real number that serves as a lower limit for the elements in that set. This means that no element in the set is less than this lower limit, providing a boundary that the elements cannot fall below. Understanding this concept is crucial for grasping related ideas like supremum and infimum, as well as recognizing the significance of the greatest lower bound property, which states that every non-empty set of real numbers that is bounded below has a greatest lower bound or infimum.
Closed interval: A closed interval is a set of real numbers that includes all numbers between two endpoints, as well as the endpoints themselves. It is denoted as [a, b], where 'a' and 'b' are the lower and upper bounds, respectively. 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.
Distance on the number line: Distance on the number line refers to the absolute difference between two points, which indicates how far apart they are from each other. This concept is crucial for understanding intervals and how to measure the size of a set of numbers. By using absolute value, we can easily determine distances regardless of direction, as distance is always a non-negative quantity.
Half-open interval: A half-open interval is a type of interval that includes one endpoint but not the other. Specifically, it can be represented as either \'[a, b)\' or \'(a, b]\', where \(a\) and \(b\) are real numbers, and the square bracket indicates inclusion while the parentheses indicate exclusion. This concept is essential when discussing various types of intervals, as it helps to define ranges of values in mathematical contexts, especially when dealing with limits or boundaries.
Interval notation on a graph: Interval notation on a graph is a mathematical way to represent a range of numbers, showing which values are included or excluded. It uses parentheses and brackets to denote whether endpoints are part of the interval, allowing for a clear visual representation of continuous sets of numbers. This notation connects to concepts like open and closed intervals, as well as how they relate to absolute values in mathematical expressions.
Number line representation: Number line representation is a visual tool used to illustrate the position of numbers along a horizontal line, which extends infinitely in both the positive and negative directions. Each point on the line corresponds to a unique real number, allowing for a clear understanding of numerical relationships such as order, distance, and operations like addition and subtraction.
Open interval: An open interval is a set of real numbers that contains all numbers between two endpoints, excluding the endpoints themselves. It is typically represented as (a, b), where 'a' and 'b' are the lower and upper bounds, respectively. Open intervals are crucial in various mathematical concepts because they signify a continuous range of values without including boundary points.
Properties of Absolute Values in Equations: The properties of absolute values in equations refer to the mathematical rules governing how absolute values behave when applied to different expressions. These properties are crucial for solving equations and inequalities involving absolute values, as they allow for the manipulation and simplification of expressions while ensuring the correct interpretation of distances from zero on the number line.
Triangle Inequality: The triangle inequality is a fundamental property of geometry and mathematics stating that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This principle not only applies to triangles but also extends to absolute values and intervals, reinforcing the idea that distances in a geometric space or on a number line maintain a specific relationship to each other.
X ∈ (a, b) means a < x < b: The notation 'x ∈ (a, b)' signifies that the value of x lies within the open interval between a and b, meaning that a is less than x and x is less than b. This concept is crucial in understanding intervals in mathematics, as it helps define the range of values that a variable can take. It establishes boundaries without including the endpoints a and b, which is essential for many mathematical applications, including limits and continuity.