Graph Simple Inequalities
So when we’re given the
inequality that’s here, it means 𝑦 is greater than or equal to one. The or equal is this bottom line
here. If we have an inequality that
is greater than or equal to, then it means that we’re going to have a straight
line, like so. If we have an inequality that’s
just greater than or less than, like this, we will have a dotty line, like this. And this is really similar as
when we had our inequalities on the number line. Remember in a number line if
it’s just greater than or less than, we have a hollow circle, and when it’s or
equal to, we have a circle that we’ve coloured in.
Now let’s try plotting 𝑦
is greater than or equal to one. So what we’ll do is, we
will pretend when we’re plotting it that it’s not just greater than or
equal to, but it is in fact just 𝑦 equals one.
So we know the line 𝑦
equals one is a horizontal line going through where 𝑦 is equal to one on
the 𝑦-axis. Now the only bit that we
need to pay attention to here is whether we’re doing a dotty or a straight
line. Well it’s or equal to, so
we’re going to do a straight line through 𝑦 equals one.
And then this is the part
that we need to focus on: is it greater or less? So you can see in this
case 𝑦 is greater than one. So what we’re going to do
is look at each side of the line. So if we look below the
line at the 𝑦-values, we can see that it is zero, minus one, and minus
two. Well that’s less than one,
so that’s not what we want. We can see above 𝑦 equals
one. We’ve got two and then
I’ll carry on going for three and four and et cetera all the way to
infinity. For they- that is greater
than one, so we will shade our wanted region, which is above the line.
So this here is where 𝑦
is greater than or equal to one. Be careful to read every
question you do with inequalities and graphs carefully because sometimes
they say shade the region or indicate the region. So you need to make sure
that you’re doing exactly what the question asks from you. In this case, we’re
shading the region that we want.
Now let’s have a go an 𝑥.
Shade the region that
satisfies 𝑥 is less than three. So we have to remember
that we’re not going to draw 𝑥 is less than three; we’re going to put onto
the graph 𝑥 equals three to help us with plotting, but the thing that we
do need to pay attention to is whether it’s a dotty or a straight line. In this case, it’s not got
an or equal to, so we’re going to be doing a dotty line.
So we need to find where
𝑥 equals three, or we can see that will be where 𝑥 is three on the
𝑥-axis, and we’ll be drawing a vertical line but that line must be dotty. As- so some places say
dashed and it honestly means exactly the same thing, so we’re looking for
where 𝑥 is less than three. So we’re gonna look on
either side of the line. Now looking at the 𝑥-axis
on the right-hand side, we can see that 𝑥 is five, well that’s greater
than three, and ten and so is that, so we don’t want greater than that we
can see on the left-hand side we’ve got zero; that’s less than three,
negative five is less than three, and so is negative ten. So we’re shading the
region that satisfies 𝑥 is less than three and that would be the left-hand
And we’re done. We have shaded the region
that satisfies 𝑥 is less than three.
Shade the region that
satisfies 𝑦 is greater than 𝑥. Again, when it comes to
plotting, what we’re actually gonna put on our graph is 𝑦 equals 𝑥.
But we need to take care
as to whether it will be dotted or straight. And in this case, we can
see it’s not or equal to, so it’s going to be dotty. We can see this is
slightly different to our previous two examples because it’s not just 𝑥
equals a constant or 𝑦 equals a constant; it’s 𝑥 equals 𝑦 or 𝑦 equals
𝑥, so when we plot it, we’re looking for every single coordinate of the
𝑥-value to be equal to the coordinate of the 𝑦.
So for example, zero and 𝑥
zero and 𝑦. Five in the 𝑥-coordinate
and five in the 𝑦-coordinate, then fi- negative five in the 𝑥 and
negative five in the 𝑦. And there we can see
negative ten and negative ten and positive ten and positive ten, so this
gives us a nice straight line; it’s gonna be exactly on forty-five degrees
with both axes.
And we must make sure this
line is dotty. So throughout this whole
line, every 𝑥-coordinate is equal to every 𝑦-coordinate. Now we’re looking for
where the 𝑦-coordinate is greater than the 𝑥-coordinate, so we’re going
to have a look above and below. So have we- if we have a
look above, we’ve got this coordinate here and this has got zero in the 𝑥
and five in the 𝑦. So in that case, five is
greater than zero, so above is going to be 𝑦 greater than 𝑥. But let’s have a look
below just so we know it anyway; I will pick this one here, so we’ve got
one two three, so we’ve got eight in the 𝑥 and then negative three in the
𝑦. Well this one 𝑥 is
clearly greater than 𝑦, so therefore looking for the top region, as we
want, where 𝑦 is bigger than 𝑥.
So in simple inequalities, we
just need to focus on plotting the graph as if it were a normal equation. Then we need to say is it
greater or less and is it dotty or straight. Those are the things we need to