As a math tutor, one question I hear often is, "What's with all these brackets and symbols?"
And I get it—when I first encountered intervals, I felt just as puzzled!
Intervals might seem like a small concept, but they’re a powerful tool in math, especially as you move into Algebra 1, 2, and Precalculus. Today, let's break down bounded intervals in a way that’s easy to understand and see why they’re so useful.
What Are Bounded Intervals?
In math, bounded intervals are ways to define a range of numbers between two points, which we call a and b. It’s like marking off a section of the number line and deciding what’s included in that section. Each type of interval tells us a little more about what numbers are “inside” or “outside” the range. Let’s go over the different types:
Closed Interval [a, b]:
This interval includes all values from a to b, including both endpoints.
We write it as [a,b], which means a≤x≤b.
I tell my students to imagine a fenced-off area with both gates closed—every number from a to b is “inside.” It’s fully inclusive, capturing both ends of the range.
Open Interval (a, b):
This interval includes all values between a and b, but not the endpoints.
We write it as (a,b) meaning a<x<b.
I compare this to keeping both gates open but not letting anyone actually stand at the boundaries. Only the numbers in between a and b belong here.
Half-Open Interval [a, b):
This interval includes a but not b.
Written as [a,b)[a, b)[a,b), it reads as a≤x<b
Think of it as one gate closed (at a) and one open (at b). It’s a handy way to keep one endpoint in the mix while keeping the other out.I find that visualizing it this way makes it clearer for my students.
Half-Open Interval (a, b]:
This includes b but not a.
Written as (a,b](a, b](a,b], it means a<x≤b
This is just the reverse of the previous interval—one end is included, and one is not.
Why Bounded Intervals Matter
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