PC - Intervals Lesson
Intervals
In order to visualize the concept of an interval, imagine cutting a section out of the real number line. The resulting section is considered an interval. Geometrically, think of an interval as corresponding to a line segment or possibly a ray. In terms of sets the real number line represents an infinite set of real numbers with infinitely many possible subsets. Thus an interval is a subset of the real numbers.
Intervals are often used to denote the solution to an inequality or a range of possible values for a given situation. Intervals come in different styles and types: Roll over each example to view the graphical representation of the example interval.
1. Open intervals do not contain their endpoint (-5, 4)
Solution:
2. Closed intervals contain their endpoints, e.g.,[-2, 8]
Solution:
3. Half-open intervals contain one but not both endpoints, e.g.,(-5, 3]
Solution:
4. Infinite intervals extend indefinitely in one or both directions, e.g.(-∞, 2], [8,∞), or (-∞, ∞)
Solution:
Interval Notation
Illustrating the different types of intervals requires a special mathematical notation for representing the set of possible real numbers taken on by a variable. This special notation, which is referred to as interval notation, uses a pair of numbers to represent the endpoints of the interval and parentheses and/or brackets to show whether the endpoints are included or excluded. For example, [-8, 4) is the interval of real numbers between -8 and 4, including -8 and excluding 4. Any point of an interval that is not an endpoint is an interior point.
Recall that infinity, ∞, and negative infinity, -∞, are NOT numbers; rather infinity is an idea of something that is endless. Thus, any interval containing infinity such as [2, ∞) is to be interpreted as one that extends indefinitely in a positive direction. Similarly, any interval containing negative infinity such as (-∞, 2] translates to one that extends indefinitely in a negative direction.
Set-Builder Notation
A collection of objects is a set, and the objects are called elements of the set. Sets are often described by listing their elements within braces, such as S = {1, 2, 3, 4, 5}. This notation is often used to describe the solution(s) to an equation. For example, the solution to |x + 4| = 2 written using set notation is {-6, -2}.
If we want to denote the set of all positive numbers less than 6, listing all the elements is impossible. In such situations using set-builder notation, T = {x|x < 6}, which is read "T is the set of all x such that x is less than 6", is an efficient way of describing the set. Capital letters are typically used to denote sets.
Putting It Together
The table below illustrates different types of intervals and the relationship between interval notation and set-builder notation.
|
Interval Notation |
Interval Type |
Set-Builder Notation |
|
(a, b) |
Open |
{x|a < x < b} |
|
[a, b] |
Closed |
{x|a < x < b} |
|
[a, b) |
Half-open |
{x|a ≤ x < b} |
|
(a, ∞) |
Infinite |
{x|x > a} |
|
[a,∞) |
Infinite |
{x|x ≥ a} |
|
(-∞, b) |
Infinite |
{x|x < b} |
|
(-∞, b] |
Infinite |
{x|x ≤ b} |
|
(-∞,∞) |
Infinite |
|
NOTE: Interval notation for an open interval looks the same as the notation used to denote an ordered pair. Knowing which way to interpret this representation will depend on the problem context. View below an interval notation video that reviews interval notation, set-builder notation, and the graphical representation of an interval.
Solving Linear Inequalities
A more common situation arises when the solution to a linear inequality such as 3x + 8 > -7 or a compound inequality such as 3x + 8 > -7 and 4 - x > 1 is needed and you must express the solution in a prescribed format. View the video below to see how to solve linear inequalities algebraically, how to use a graphing calculator to solve linear inequalities, and how to express the solution to a linear inequality using inequalities and interval notation.
Solving Absolute Value Equations and Inequalities
Applying the piecewise definition of absolute value is frequently required in calculus.
View the book below to review how to find graphical solutions of absolute value equations and inequalities.
Thinking of absolute value as distance from 0 on a number line supports the formal definition since we know distance from 0 is always positive. Don't be confused when you see the negative sign in the second part of the piecewise definition. If you think of the negative sign as meaning the opposite of the expression whenever the expression is negative, then the result will be positive.
- Click HERE to view the video below to investigate how to solve |2x- 1| < 3 algebraically. Links to an external site.
- Click HERE to view the video below to explore how to solve |2x1| > 3 algebraically. Links to an external site.
View a video below on solving absolute value inequalities.
Intervals Practice
Check your skills below by matching the solution with the problem.
Intervals: Even More Problems!
Complete additional problems from your textbook and/or online resources as needed to ensure your complete understanding of interval-related content.
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