WEBVTT
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Graph Two-Variable Inequalities
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There, given this inequality, we can say this is, 𝑦 is greater than or equal to three 𝑥 minus one.
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So how we will plot this is, we’ll basically pretend that we’re plotting instead.
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And if we were plotting the line 𝑦 equals three 𝑥 minus one.
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First thing we would need is a table of values.
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Well we need to pick three coordinates.
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My favourites is negative one, zero, and one.
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It can be more than three values, but it can never be less.
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So we’re going to take each of these 𝑥-coordinates that we’ve chosen now and substitute them into the function to get the 𝑦-value.
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So for the first one, we put 𝑦 equals three multiplied by negative one minus one.
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Well three multiplied by negative one is negative three.
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And subtracting one from that, we get negative four.
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Then having a go for zero, we know that three multiplied by zero is zero.
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So then zero minus one is minus one.
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And then finally, substituting in one on the 𝑥-coordinate, we’ll have three multiplied by one, which is three, and minus one is two.
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So now we have some coordinates.
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Those coordinates are: Negative one, negative four; zero, negative one; and one, two.
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We’ve got each of these coordinates by obviously looking at the 𝑥-coordinates in the table of values and looking at the 𝑦, and then saying, well that’s the 𝑥 and that’s the 𝑦, to find each of the set of coordinates.
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Now what we’ll need to do is plot this graph.
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But this is where we need to care about it being an inequality again.
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So plotting each of these points we’ve got negative one, negative four; zero, negative one; and one, two.
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Now this is an inequality that is an “or equal to” inequality, as we can see here.
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So what that means is, when we come to plot the graph we must have a straight line, rather than a dotty line.
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Because for dotty line, it’s just greater than or less than, whereas with a straight line, we also include “or equal to”.
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And this follows, if you remember putting inequalities on a number line, we put a hollow circle in the case where we had just greater than and or less than, and a coloured-in circle for “or equal to”.
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So anyway, going back to the question, we can see ours is an “or equal to”, so it needs to be a straight line.
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This, that I did earlier.
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Now the thing with two-variable inequalities, the thing that’s different from when we have simple linear inequalities, is we can’t just say “Oh okay that means I’m going to shade this side”.
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We need to actually do some testing.
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So we’re going to test above and test below, and see which fits our inequality.
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So if we test this point below of five, five and we’ll test negative five, ten above.
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We can really pick any points.
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But we just need to see, do these points satisfy the inequality.
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So we’ve got that 𝑦 is ten, so we put ten is greater than or equal to three multiplied by the 𝑥-value, which is negative five, minus one.
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Well we know that three multiplied by negative five is negative fifteen, and that negative fifteen minus one is negative sixteen.
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So ten is greater than or equal to negative sixteen.
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So above works.
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But we’re not just going to take that as an answer and say “Well there we go.
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I’m going to shade the region above, as that satisfies the inequality” because we have to assume that we always might’ve done something wrong.
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So let’s check our test below, and it should be wrong; it should not satisfy the inequality.
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So we’ve got five is greater than or equal to three multiplied by five minus one.
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Well three multiplied by five is fifteen, and subtracting one from that we get fourteen.
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Now it’s not true that five is greater than or equal to fourteen.
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So we know that below doesn’t work, which is exactly what we wanted.
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So we can shade the region that satisfies the inequality, and that is above.
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And now we have shaded the correct inequality by plotting the line and testing above and testing below.
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Now one thing that has to be in place for us to be able to do a table of values is that our inequality has to be in the form 𝑦 equals 𝑚𝑥 plus 𝑐, which we can see that it doesn’t in this question.
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So in this question, we need to first get our inequality into the form 𝑦 equals 𝑚𝑥 plus 𝑐.
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So what we essentially need to do is, rearrange to get 𝑦 the subject of the function.
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So what we’re gonna do first, is subtract two 𝑥 from both sides.
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And that gives us on the left-hand side three 𝑦, which is less than or equal to seven minus two 𝑥 on the right-hand side.
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We can see that this still isn’t quite in the way that we need it.
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We need it to be just 𝑦, is it less than or equal to.
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So we’re going to have to divide both sides by three, giving us 𝑦 is less than or equal to seven minus two 𝑥, all divided by three.
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And although this isn’t exactly in the form that we want it, we can see that it is some number and some coefficient of 𝑥 on the right-hand side with 𝑦 by itself on the left.
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So we can use this to have a table of values and plot the graph.
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So then substituting in negative one to the function, we get 𝑦 is less than or equal to seven minus two times negative one all divided by three.
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The negatives on the top become a positive, so we’ve got seven add two, which is nine, divided by three, just gives us an answer of three.
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And then substituting in zero and one, we get some fractions of seven over three and five over three.
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So now this gives us some coordinates of negative one, three; zero, seven over three; and one, five over three.
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These aren’t the nicest coordinates to plot, but we can definitely use them.
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And when we do plot our graph, we get this.
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And then finally, we must test above and test below.
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So we’ll choose five, five above and negative five, negative five below.
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So to make it easier for us, let’s use the original function.
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Well we’ve got three multiplied by five, which is fifteen.
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Plus two multiplied by five, which is ten.
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And that must be less than or equal to seven, which is not.
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So the above doesn’t work.
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So we need to test below.
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Now again, using the original function, we’ve got three multiplied by negative five, which is negative fifteen.
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Plus two multiplied by negative five, which is negative ten.
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And that is less than or equal to seven.
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So below works.
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So to shade the region that satisfies the inequality, we need to shade below the line.
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And now we have it.
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We have finished the inequalities with two variables.
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And the most important things are to remember to plot using a table of values, and to check above and below the line, to be able to know what side to shade.