State the system of inequalities
whose solution is represented in the following graph.
And we’ve been given a graph with
two lines; let’s call them one, which is the dotted line, and two, which is the
solid line. Now for region one, we’re talking
about points in the red region here, so above that line. And for region two, we’re talking
about this yellow region here, so points above this line. And then this region I’ve shaded
here is the intersection of those two regions.
So with two lines and two
corresponding regions, we’re looking for two inequalities. And for the first one, we’ve got a
dotted line. So points on the line are not
included in the region if you got a dotted line.
And that implies that we’re talking
about a strict inequality, so 𝑦 is greater than something or 𝑦 is less than
something. And because we’re looking at points
which are above that line, in fact, in our case we’re gonna have a
Well first of all, let’s work out
the equation of that line. And to do that, we need to know the
slope and the 𝑦-intercept. So if I take a point on that line
and then I increase the 𝑥-coordinate by one, the corresponding 𝑦-coordinate on the
line would decrease by three.
And that happens all the way down
the line; increase the 𝑥-coordinate by one, the 𝑦-coordinate goes down by three;
increase the 𝑥-coordinate by one, the 𝑦-coordinate goes down by three. So the slope is negative three. Now we need to think about the
𝑦-intercept. And that means where does it cut
the 𝑦-axis. When my 𝑥-coordinate is zero,
what’s the corresponding 𝑦-coordinate? Well in this case, it’s negative
So my 𝑦-intercept is negative
eight. Now the general equation of a
straight line is 𝑦 is equal to the slope times 𝑥 plus the value of the
And with a slope of negative three
and a 𝑦-intercept of negative eight, I can put those values into my general
equation and I’ve got 𝑦 is equal to negative three 𝑥 plus negative eight and would
normally write that as 𝑦 equals negative three 𝑥 take away eight.
Now that equation tells us about
all of the 𝑦-coordinates on that line, and we’re interested in that red region and
all the points which are in fact above the line not on it. So the 𝑦-coordinates of points in
this region up here are all greater than the values on the line.
So for instance when 𝑥 equals
zero, point on the line has a 𝑦-coordinate of negative eight. But we’re in our region; we’re
interested in all these 𝑦-coordinates all the way up here, which are greater than
So we can represent that red region
with this inequality. All the 𝑦-coordinates are greater
in the red region than their 𝑥-coordinate times negative three and then take away
eight from that.
Okay, let’s go and work out the
inequality for the yellow region. Well equation two has got a solid
line, and that tells us that points on the line are included in the region.
So we’re looking for an inequality
which is something like 𝑦 is greater than or equal to something or 𝑦 is less than
or equal to something. And again in this particular case,
all the points that we’re interested in are above that line; the yellow region is
above the solid line , so it’s gonna be 𝑦 is greater than or equal to
So let’s go on and work out the
equation of that solid line. So again we need to work out the
slope and the 𝑦-intercept. And for the solid line, every time I increase my
𝑥-coordinate by one, my 𝑦- coordinate goes up by three. So my slope is positive three.
And the 𝑦-intercept, when the
𝑥-coordinate is zero, the corresponding 𝑦- coordinate is three; that’s where it
cuts the 𝑦-axis. So the 𝑦-intercept is also
And again, we can now use that
information to write out the equation of that second line. With a slope of positive three,
we’ve got 𝑦 equals three 𝑥 and the 𝑦-intercept is also positive three. So the equation representing points
on that solid line is 𝑦 equals three 𝑥 plus three.
So for that particular line, for
that equation, if 𝑥 was equal to one, 𝑦 on the line would be equal to six. But our region not only includes
that point, it includes all the points above it in the region. So 𝑦 can be equal to three 𝑥 plus
three, but it can also be greater than that.
𝑦 is greater than or equal to
three 𝑥 plus three. So points in the red region follow
this inequality; points in the yellow region follow this inequality; and points in
this region here follow both of those inequalities.
So our answer consists of both of
those inequalities: 𝑦 is greater than three 𝑥 plus three and 𝑦 is greater than
negative three 𝑥 take away eight.