APH Graph Board, Part 6, Graphing a system of inequalities

A TSBVI Outreach Tutorial.
APH Graphic Aid for Mathematics
a.k.a. Graph Board.
Part 6: Graphing a system of inequalities using the boundary lines:
y = (1/2)x + 3 and y = -2x + 3
Presented by Susan Osterhaus,
a Texas School for the Blind and Visually Impaired Outreach Math Consultant.
Okay, we are gonna go back to our system of equations
and now we're going to work on a system of inequalities
which means let's go back to all that shadings and so forth.
So, I'm gonna leave it, I'm gonna leave these boundary lines there.
But if we talk about y is less than or equal to (1/2)x + 3.
Remember this was the y equals (1/2)x + 3, so, less than that, would be everything down here.
So the students would be shading or putting a piece of paper
or putting their hand down here.
So, that's this portion of the graph.
And, now I'm going to say that we're going to have
y is greater than or equal to -2x + 3 which is this graph. Okay.
So, for this one, it's all of this and for this one,
I'm saying greater than or equal to.
So, that's this area.
And, some... I will have my students actually do this.
And then, I will say to them, "Okay, where do your arms overlap?"
Okay, not, not right down in here.
Not way up here but actually right in this area is where they overlap
and that is you actual solution in this case.
So again, they are doing something like this and realizing that this is the area.
Now, once that they become experienced, I'm not saying that they have to,
you know, totally do all that, they can conceptualize this in their head,
but at the beginning, that whole thing of actually putting your arms over there,
sometimes I will come over if they can't get it.
I will have them put my, their arm there and then I'll put my arm
and then they can really tell the difference between what's their arm and what's mine
and where they're overlapping.
And in this case, this would be our solution.
Including all of these boundary points.
So again, this would be shaded in or we would put in our pushpins.
Let's just, we'll go ahead and put the pushpins in there
just because I think that might be a little easier for your benefit right now.
And by the way, we do have a graphing calculator that will do this and do this wonderfully.
Except, it doesn't do the shading as of yet.
So, your students are still gonna have to manually do the shading
even if they use the audio graphing calculator for instance.
But anyway, this way they have an idea of exactly where the solution set is.
Now, how we can change this if we take off the equal part,
we could change all this if we still leave it as y is less than (1/2)x + 3 and take this off.
Oops, if I can get these off.
This is the bad thing of putting too many pushpins on.
Now, let's just leave it like that and leave the other one.
That means, the solution set are all of these points,
not these boundary points, and then again, everything inside.
But we're gonna change this and make it not y is greater than or equal to
but just y is greater than -2x + 3.
Just lost a point.
By the way, the reason that point just came out,
was because this board has been used multiple times and after awhile,
it gets punched a little too much.
So, anyway, what we've got here is now, again the area is, this is like the boundary
but these points are not included, but everything inside...
Sometimes the students have a little bit of trouble about thinking
about boundary and I tell them, "This is kind of like a wall,
that's been painted, newly painted."
And you can go up as close as you can to that wall but you're not allowed to touch it
because you'll get paint all over yourself.
So, think of these as the boundary lines but you can get almost next to it
but not quite there because if you touch, you'll get wet with paint.
So again, this is the solutionary and not these points, not these points,
but everything inside all of this is in the solution set.
And, I've tried to show you various ways to do that and I'm trying to think a minute here.
You can make this even more complicated and if it does,
if you end up with three or four inequalities
you'll get into something called linear programming.
And, you're still able to do it with this and a lot of times with linear programming,
where everything crosses, and it's a little bit difficult to see that right now.
Let me take off a few pins and see if I can kind of show you something like that.
You might end up...
Let me take some pushpins off here.
It's where they cross here and if you used...
Oops. Let me see.
We've got this.
And now, actually, I'm going to be...
I'm not going to be using, I won't be using this particular equation anymore.
But you can see if we use the Y-axis, the X-axis and this particular line.
We would actually form a triangle
and that might be what's used in linear programming
and what the area in this case, would be the solution,
would be right what's inside that triangle.
In this particular case in the second quadrant.
But, it might also be something like... Let me try this one.
If we kept that line, it might be something...
It's gonna be very, very difficult to see that it might be something included
right in, right in that very, very small area inside.
This is a little bit better to see it.
However, a lot of times with linear programming
it's gonna be basically up here in your first quadrant.
So, this is not really a terrific example of that.
But you can do as difficult a thing as linear programming with the graph board.
Just to let you know that.
**Captions by Project readOn**