Math Lesson 14.5.1 - Recalling Intervals and Segments (and their Combinations). Showing Intervals and Segments Visually on a Number Line.

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Welcome to our Math lesson on Recalling Intervals and Segments (and their Combinations). Showing Intervals and Segments Visually on a Number Line. , this is the first lesson of our suite of math lessons covering the topic of Line Segments, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Recalling Intervals and Segments (and their Combinations). Showing Intervals and Segments Visually on a Number Line.

In tutorial 10.1, when dealing with number sets that may be solutions for inequalities, we have provided the following information about intervals and segments:

"There are some special symbols that represent sets of numbers determined by inequalities#2. Let's explain them through the following table:

The above table shows the algebraic (symbol) representation of intervals and segments (and combinations derived from them). However, there is another method of showing the above number sets visually. Thus, we use the following symbols for showing all the above sets in a number line:

1. The figure below shows a segment in two ways: in algebraic notation, i.e. [-2, 6], and visually, on a number axis, where the endpoints are marked by black (filled) dots. The segment in question represents the number set that includes all values from -2 to 6, including these endpoints.
2. The figure below shows the interval from -1 to 5 in the two ways mentioned above. When showing visually on a number axis, the endpoints of an interval are marked by empty dots. Therefore, the interval in question represents the number set that includes all values from -1 to 5 but not these endpoints.
3. The figure below shows the half-segment from -3 to 2 in two ways: in algebraic notation, i.e. [-3, 2], and visually, on a number axis. Therefore, the half-segment in question represents the number set that includes all values from -3 to 2 without the right endpoint (without 2). Therefore, the half-interval in question represents the number set that includes all values from -5 to 6 without the left endpoint (without -5).
4. The figure below shows the half-segment from -2 to plus infinity in two ways: in algebraic notation, i.e. [-2, + ∞), and visually, on a number axis. The half-segment in question represents all values on the right of -2, including this endpoint. The arrow indicates extension to infinity in that direction (due right).
5. The figure below shows the interval from 0 to plus infinity in two ways: in algebraic notation, i.e. (0, + ∞), and visually, on a number axis. The interval in question represents all values on the right of 0, without including this endpoint. The arrow indicates extension to infinity in that direction (due right).
6. The figure below shows the half-segment from minus infinity to 4 in two ways: in algebraic notation, i.e. (-∞, 4], and visually, on a number axis. The half-interval in question represents all values on the left of 4, including this endpoint as well. The arrow indicates extension to infinity in that direction (due left).
7. The figure below shows the interval from minus infinity to 2 in two ways: in algebraic notation, i.e. (-∞, 2), and visually, on a number axis. The interval in question represents all values on the left of 2, without including this endpoint. The arrow indicates extension to infinity in that direction (due left).
8. The figure below shows the interval from minus infinity to plus infinity in two ways: in algebraic notation, i.e. (-∞, + ∞), and visually, on a number axis. The interval in question represents all values of the number axis, without exception.

Remark! You may have noticed that when a number set contains both the symbols of interval and segment, it takes the name "half-(the set symbol on the left endpoint)". For example, the script (a, b] reads "the half-interval from a to b", while the script [a, b) reads "the half-segment from a to b."

Example 1

Write in symbols the number sets indicated by the figures below.

Solution 1

1. The first figure shows a half-interval, as the left endpoint contains a white (blank) dot, while the right one contains a black (filled) dot. Therefore, we write this number set as (-7, 8], as every division in the number line represents 1 unit.
2. The second figure shows a segment, as both endpoint contain black (filled) dots. Therefore, we write this number set as [-28, 34], as every division in the number line represents 2 units.

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