Graphing: Why one is part, and one is not.

[John Whitaker]

Graph: x < 2
The solution of x < 2 are all those numbers less than 2. They are shown on the graph by shading all points to the left of 2. The open circle at 2 indicates the 2 is not part of the graph. (So states my book.) It does not say WHY 2 is NOT part of the graph.

The next item states: The solutions of x > or = to –3 are shown on the number line by shading the point from –3 and all points to the right of –3. The closed circle at –3 indicates that –3 IS part of the graph (but, as before, it does not say WHY this distinction is made). The first one IS, and the second IS NOT. Why? Can you clear this up for me? Thank you.
John Whitaker

 

[soroban]

Hello, John!

Graph: \(\displaystyle \,x\,<\,2\)
The solution of \(\displaystyle x\,<\,2\) are all those numbers less than 2.
They are shown on the graph by shading all points to the left of 2.
The open circle at 2 indicates the 2 is not part of the graph.
It does not say WHY 2 is NOT part of the graph.

They assume you understood what "less then 2" means.

What numbers are less than 2?
Of course, 1, 0, -1, -2, . . . but also 1.9 and 1.99 and 1.999 ... etc.

Is 2 less than 2? . . . No, 2 is equal to 2.
So 2 is not a number which is less than 2.

And that's why it's not part of the graph . . .

The next states: The solutions of \(\displaystyle x\,\geq \,-3\) are shown on the number line
by shading the point at –3 and all points to the right of –3.
The closed circle at –3 indicates that –3 IS part of the graph
(but, as before, it does not say WHY this distinction is made).

If \(\displaystyle x\,\geq\,-3\) (\(\displaystyle x\) is greater than -3 or equal to -3),
. . then we want all the points "above" -3 . . . and -3 itself.
Doesn't that make sense?

 

[John Whitaker]

It makes perfect sense. I think Bittinger could have mentioned it. Thank you.