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Graph Simple Inequalities
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So when we’re given the inequality that’s here, it means 𝑦 is greater than or equal to one.
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The or equal is this bottom line here.
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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.
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If we have an inequality that’s just greater than or less than, like this, we will have a dotty line, like this.
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And this is really similar as when we had our inequalities on the number line.
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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.
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Now let’s try plotting 𝑦 is greater than or equal to one.
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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.
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So we know the line 𝑦 equals one is a horizontal line going through where 𝑦 is equal to one on the 𝑦-axis.
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Now the only bit that we need to pay attention to here is whether we’re doing a dotty or a straight line.
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Well it’s or equal to, so we’re going to do a straight line through 𝑦 equals one.
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And then this is the part that we need to focus on: is it greater or less?
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So you can see in this case 𝑦 is greater than one.
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So what we’re going to do is look at each side of the line.
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So if we look below the line at the 𝑦-values, we can see that it is zero, minus one, and minus two.
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Well that’s less than one, so that’s not what we want.
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We can see above 𝑦 equals one.
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We’ve got two and then I’ll carry on going for three and four and et cetera all the way to infinity.
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For they- that is greater than one, so we will shade our wanted region, which is above the line.
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So this here is where 𝑦 is greater than or equal to one.
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Be careful to read every question you do with inequalities and graphs carefully because sometimes they say shade the region or indicate the region.
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So you need to make sure that you’re doing exactly what the question asks from you.
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In this case, we’re shading the region that we want.
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Now let’s have a go an 𝑥.
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Shade the region that satisfies 𝑥 is less than three.
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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.
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In this case, it’s not got an or equal to, so we’re going to be doing a dotty line.
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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.
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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.
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So we’re gonna look on either side of the line.
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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.
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So we’re shading the region that satisfies 𝑥 is less than three and that would be the left-hand side.
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And we’re done.
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We have shaded the region that satisfies 𝑥 is less than three.
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Shade the region that satisfies 𝑦 is greater than 𝑥.
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Again, when it comes to plotting, what we’re actually gonna put on our graph is 𝑦 equals 𝑥.
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But we need to take care as to whether it will be dotted or straight.
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And in this case, we can see it’s not or equal to, so it’s going to be dotty.
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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.
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So for example, zero and 𝑥 zero and 𝑦.
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Five in the 𝑥-coordinate and five in the 𝑦-coordinate, then fi- negative five in the 𝑥 and negative five in the 𝑦.
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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.
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And we must make sure this line is dotty.
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So throughout this whole line, every 𝑥-coordinate is equal to every 𝑦-coordinate.
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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.
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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.
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So in that case, five is greater than zero, so above is going to be 𝑦 greater than 𝑥.
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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.
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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.
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So in simple inequalities, we just need to focus on plotting the graph as if it were a normal equation.
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Then we need to say is it greater or less and is it dotty or straight.
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Those are the things we need to focus on.