Lesson video

Hi everyone, Ms. Jones here.

Today is our final part of solving inequalities graphically.

And so that is what we're going to be focusing on today.

But before we can start, pause the video here to make sure you have a pen and some paper, and you'll probably also need a ruler, if you have one, as well as removing any distractions and trying to find a nice quiet space to work.

Pause the video to make sure you've got all of that ready so that we can begin.

Okay, let's start.

So the first thing I would like you to do is find the points of intersection of the following lines.

Y equals three X add seven, Y equals seven X add three, and Y equals three subtract X.

So you're going to need to draw these out nice and accurately for me on a set of axes and find the points of intersection.

Pause the video to do this.

Now this is what you should have, it should have looked like.

It's really important that even if you struggle to draw that graph, you've got that graph drawn at somewhere now because we will be using that later on in the lesson.

So the points of intersection you should have found is here at A, where we've got the point of intersection of three X add seven and Y equals three subtract X.

And that point of intersection is negative one four.

At point B, where we've got Y equals three subtract X intersecting with Y equals seven X add three, to find zero three, and point C, which is Y equals three X add seven and Y equals seven X add three, where they intersect at one 10.

Really well done if you managed to find all three of those.

Brilliant job So we can solve inequalities using graphs, which we've seen before, but this time it's a little bit different, 'cause we have two lines now.

We've got Y equals three X add seven and Y equals three subtract X.

So it's two lines taken from the previous task.

And we would like to solve three X add seven is greater than three subtract X.

So you've drawn both of these lines, and similar to what we would do if we were to solve, if this was an equation rather than inequality, we find the point of intersection.

And the point of intersection is at X equals negative one.

We know at that point those two lines are equal, but we want to know at what points three X add seven, that line, is greater than three subtract X.

And we can do that by looking at it.

So the three X add seven line is, is this one here.

Pretend that's drawn nicely along that line.

And this is the one we want to know where it's bigger at, where the Y values are bigger at.

And we can see that the Y values are bigger here.

And obviously that continues infinitely.

And so we are looking at this side of X equals negative one.

So we are looking at the region here.

X is greater than negative one, and that is our solution.

That is our inequality.

And you can see if we were to solve this algebraically, that is what we would have gotten as well.

But remember graphically is just as important, and quite often easier if we were given the graph already.

So we can see the solution is X is greater than negative one.

If we were to have the inequality sign the other way around, we would be looking for where three X add seven is less than three subtract X.

And we'd be looking for these Y values here, where it's less, it's lower down, that three X add seven is lower down than the three subtract X on the Y axis.

So it would be looking at X is less than negative one.

Pause the video now, to you complete your independent task, which will be on the next slide.

So the first question asks you to draw two lines, Y equals negative two X add four and Y equals X add one, and it then asks you to find the point of intersection and then use that to solve negative two X add four is greater than X add one.

So we are going to look where those lines are, where the point of intersection is, and where negative two X add four, that line, is greater than X add one.

And we can see here, we've got our point of intersection here, X equals one, but where negative two X add four has got greater Y values is actually where X is less than one, so this side here, because this is the line we are looking for to be greater.

So X is less than one.

For the second question, it's very similar.

Draw Y equals three X add four, and X add Y equals negative four on a set of axes, and find the point of intersection, which we've got here.

And then we are using the graph to solve three X add four is less than negative X subtract four.

Now hopefully you saw the link between negative X subtract four, and this equation here.

If you were to rearrange this equation, you would have Y equals negative X subtract four.

That's why we can use that graph to solve this inequality.

And we can see where three X add four, which is this line here, is less than this line here, is all of the points here, which will continue forever, so it's X will be less than negative two.

And when you think of it, if that sign changed, this sign would change to that, okay.

And then question three is just straight.

Use a graph to solve negative four X add three is greater than or equal to negative two X subtract one.

So this time you needed to draw your lines on your axes of Y equals negative four X add three, and Y equals negative two X subtract one, and find where they intersect.

And again, you can see that the symbols, if it's got greater than or equal to, it's going to be something or equal to there.

And that's those ones.

So really well done if you got the first two correct, and an extra special well done if you got the third one correct, which was just asking you to go for it on your own without those kinds of steps.

So really great job there, well done.

Your final task, and I did say you'd need to use that graph from the try this again, and I was right.

Use your graph of Y equals three X add seven, Y equals seven X add three, and Y equals three subtract X from the try this task to find the range of solutions for which, and then we've got four questions using different lines.

So we need to think back to that graph, find that graph on your piece of paper, and use the Y equals seven X add three, and the Y equals three subtract X line, find the intersection of those lines, and find where they're greater than, or less than each other.

This one's a little bit trickier because we've got three lines.

So try and block out that extra line that we don't need for each question so that we can really see where we're looking and which solution is going to be right for which graphs.

Pause the video here to have a go at this.

And here are your solutions.

So for the first one, seven X add three, which was this line here, and three subtract X, which is this line here, we can see that seven X add three is going to be greater than three subtract X, are all the values where that X is greater than zero, 'cause we got the point of intersection at X equals zero, and then we're looking at all the values in this direction.

And similarly, for the other ones.

So really, really well done if you got those right, it can be quite tricky to separate off those lines, you may have found that you need to draw another graph and that's fine.

And well done if you checked your answers by solving them algebraically, that's exactly the kind of thing that you should be doing, so that's brilliant, well done if you did that.

And that brings us to the end of the lesson and the end of solving inequalities graphically.

So absolutely fabulous job and make sure you take the quiz to check your understanding.

It's been great teaching you this little bit of the topic.

So really, really well done.