Which regions on the graph contains
solutions to both of the following inequalities, 𝑦 is greater than 𝑥 and 𝑦 is
less than or equal to two 𝑥 minus four?
To solve this problem, we’re
actually going to start with this inequality that says that 𝑦 is greater than
𝑥. So as you can see on our graph,
we’ve already got the line 𝑦 equals 𝑥. Well we’re interested in all the
regions where 𝑦 is greater than 𝑥. So we actually want everything
that’s this side of the line. So therefore, we can eliminate all
these regions as these regions are all where 𝑦 is less than 𝑥.
Okay, great! Let’s move on to the next inequality. Well our next inequality states that 𝑦 is less than or equal to two 𝑥
minus four. And what I wanna do actually just
to see which one of the lines on our graph represents two 𝑥 minus four.
I’m actually gonna substitute some
values in. The values that we’re gonna
substitute in for 𝑥 are 𝑥 is equal to zero, two, and four. And I am gonna put those values in
just so we’ve got enough to show exactly which line is going to be equal to our 𝑦
is equal to two 𝑥 minus four.
Well first of all, if I substitute
in 𝑥 is equal to zero, I’m gonna get 𝑦 is equal to zero minus four. So therefore, 𝑦 is gonna be equal
to negative four. So then our first point will be
zero, negative four. So I’ve now marked that on the
graph so we can see where that is.
Okay, substituting our next value
of 𝑥, so 𝑥 is equal to two, we get 𝑦 is equal to two multiplied by two minus
four, which actually give a 𝑦-value of zero. So now we can mark that on our
graph. Okay, great! That’s now on there so it looks
like we’ve identified the line that’s going to be 𝑦 is equal to two 𝑥 minus
four. But we’ll just do our final value
just to make sure.
So in substituting, 𝑥 is equal to
four. So we get 𝑦 is equal to two
multiplied by four minus four, which is equal to four. So great! Our final point is gonna be at the
coordinates four, four. Okay, with that marked on the
graph, it’s clear that we’ve now identified our line, 𝑦 equals two 𝑥 minus
So now we’ve got 𝑦 is equal to two
𝑥 minus four. We can actually go back to our
inequality. And what we want, we want it when
𝑦 is less than or equal to two 𝑥 minus four. So therefore, we can eliminate all
the regions that are actually going to be greater than two 𝑥 minus four.
So we’re left with one region. And that region is 𝐺. So we can say that region 𝐺
contains solutions to both of our inequalities. And yes, we’ve actually reached our
final answer. But I just want to double
check. So I want to bring your attention
to our inequality signs.
For our first inequality, it says
that 𝑦 is greater than 𝑥. So therefore, it’s just greater
than. So it’s not greater than or equal
to. So what we want is a dashed
line. And if we actually take a look at
the line 𝑦 equals 𝑥, yes we can see it’s a dashed line. So that’s great!
And for our second inequality, it
says that 𝑦 is less than or equal to two 𝑥 minus four. And because it’s less than or equal
to, we actually want a solid line. So look at the line. Yes! Great! It’s a solid line. So yep, we can be happy that region
𝐺 contains solutions to both our inequalities.