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 side.
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 focus on.