Systems of inequalities
Graphing systems of inequalities 2 Graphing systems of inequalities 2
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- We're asked to determine the solution set of this system,
- and we actually have three inequalities right here.
- A good place to start is just to graph the solution sets for
- each of these inequalities and then see where they overlap.
- And that's the region of the x, y coordinate plane that
- will satisfy all of them.
- So let's first graph y is equal to 2x plus 1, and that
- includes this line, and then it's all the points greater
- than that as well.
- So the y-intercept right here is 1.
- If x is 0, y is 1, and the slope is 2.
- If we move forward in the x-direction 1, we move up 2.
- If we move forward 2, we'll move up 4, just like that.
- So this graph is going to look something like this.
- Let me graph a couple more points here just so that I
- make sure that I'm drawing it reasonably accurately.
- So it would look something like this.
- That's the graph of y is equal to 2x plus 1.
- Now, for y is greater than or equal, or if it's equal or
- greater than, so we have to put all the region above this.
- For any x, 2x plus 1 will be right on the line, but all the
- y's greater than that are also valid.
- So the solution set of that first equation is all of this
- area up here, all of the area above the line, including the
- line, because it's greater than or equal to.
- So that's the first inequality right there.
- Now let's do the second inequality.
- The second inequality is y is less than 2x minus 5.
- So if we were to graph 2x minus 5, and something already
- might jump out at you that these two are
- parallel to each other.
- They have the same slope.
- So 2x minus 5, the y-intercept is negative 5. x is 0, y is
- negative 1, negative 2, negative 3, negative 4,
- negative 5.
- Slope is 2 again.
- And this is only less than, strictly less than, so we're
- not going to actually include the line.
- The slope is 2, so it will look something like that.
- It has the exact same slope as this other line.
- So I could draw a bit of a dotted line here if you like,
- and we're not going to include the dotted line because we're
- strictly less than.
- So the solution set for this second inequality is going to
- be all of the area below the line.
- For any x, this is 2x minus 5, and we care about the y's that
- are less than that.
- So let me shade that in.
- So before we even get to this last inequality, in order for
- there to be something that satisfies both of these
- inequalities, it has to be in both of their solution sets.
- But as you can see, their solutions sets are completely
- non-overlapping.
- There's no point on the x, y plane that is in both of these
- solution sets.
- They're separated by this kind of no-man's land between these
- two parallel lines.
- So there is actually no solution set.
- It's actually the null set.
- There's the empty set.
- Maybe we could put an empty set like that, two brackets
- with nothing in it.
- There's no solution set or the solution set of
- the system is empty.
- We could do the x is greater than 1.
- This is x is equal to 1, so we put a dotted line there
- because we don't want include that.
- So it would be all of this stuff.
- But once again, there's nothing that satisfies all
- three of these.
- This area right here satisfies the bottom two.
- This area up here satisfies the last one
- and the first one.
- But there's nothing that satisfies both these top two.
- Empty set.
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