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So let's try to graph the set of pairs x,y satisfying 2x + y
greater than or equal to 4,
2x - 3y less than or equal to 6,
x and y are both nonnegative.
How do you handle this?
First of all,
there are
a couple of easy inequalities.
If you look at the last two,
x greater than or equal to 0, y greater than or equal to 0,
those are relatively easy to draw.
So we'll work with those first.
x greater than or equal to 0 means that all the points with
x value greater than or equal to 0
and so they are all the points
on the right of the y-axis.
So that's x greater than
or equal to 0.
Now, what about y greater than or equal to zero?
It's everything on or above the x-axis.
And that's y greater than or equal to 0.
So immediately we know that we are looking at points on the first quadrant, right here.
Now what about these two other inequalities: 2x + y greater than or equal to 4,
and 2x - 3y less than or equal to 6.
Well, less focus on something that we know how to do.
For example, if we
look at the equation instead of the inequality; 2x + y = 4,
that defines a line.
And drawing a line is not that difficult. We just need to find two
points on the line.
The line is through those two points.
So if I ask you instead to draw 2x + y = 4,
well what does it look like?
Well, we can find two points by looking at where it intersects the axes.
For example,
it we set y = 0,
then will know where the line intersects the x-axis.
So when y = 0,
x must be 2.
Remember we're looking at uh... we're trying to draw the line
2x + y = 4. So when y is zero,
x is 2. And when
x is 0,
y is 4.
So immediately we know that the line
defined by 2x + y = 4 has to go through the two points
(2,0) and (0,4).
So (2,0) is right here
and (0,4) is right here. So we just need to draw a line through these two points.
Let me just move this over.
So
the line crosses the two points.
Now, this is the line for 2x + y = 4. But we want 2x + y greater than or equal to 4.
So not just points on the line but points
that are somehow
on the side of the line that gives 2x + y greater than or equal to 4.
So how do we figure that out?
One trick is to test a point.
If we look at the origin, (0,0)
well, it's not
going to be on the side of the line that satisfies 2x + y greater than or equal to 4
because if you plug in x = 0, y = 0, you get 0. And 0 is not greater than
or equal to 4. It is less than 4.
So the origin can not be on this side
defined by 2x + y greater than or equal to 4.
So that means
everything else that is on the same side as the origin
cannot be a point on the side defined by this inequality.
And therefore, we must have points
over on this side of the line.
Now by the same reasoning, we can draw 2x - 3y
less than or equal to 6.
So let's first draw 2x - 3y = 6.
That's just a line.
And when y is zero,
x is three.
When x is 0,
y
is -2.
All right. So
we have to draw the line through (0,-2) and
(3,0).
It's going to look something like this.
Again, this is the line defined by 2x - 3y = 6.
But we want
the points
satisfying the inequality 2x - 3y
less than or equal to 6.
Again we can use our trick. Let's see.
Well, if you plug in (0,0), what do we get?
Well, 2 times 0 is 0 and 3 times 0 is 0, so
0 is less than or equal to 6.
So that means 0
is a point on the side
defined by 2x - 3y
less than or equal to 6.
So everything that is
on the same side
of the origin with respect to this line
will satisfy this inequality.
And so
we can mark such points
by these two arrows on this line.
Now combining what we have drawn in,
the set of points satisfying all four inequalities is going to be given by
this
blue shaded region.
And that's our answer.