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We're asked to determine the solution set of this system,

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and we actually have three inequalities right here.

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A good place to start is just to graph the solution sets for

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each of these inequalities and then see where they overlap.

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And that's the region of the x, y coordinate plane that

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will satisfy all of them.

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So let's first graph y is equal to 2x plus 1, and that

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includes this line, and then it's all the points greater

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than that as well.

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So the y-intercept right here is 1.

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If x is 0, y is 1, and the slope is 2.

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If we move forward in the x-direction 1, we move up 2.

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If we move forward 2, we'll move up 4, just like that.

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So this graph is going to look something like this.

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Let me graph a couple more points here just so that I

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make sure that I'm drawing it reasonably accurately.

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So it would look something like this.

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That's the graph of y is equal to 2x plus 1.

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Now, for y is greater than or equal, or if it's equal or

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greater than, so we have to put all the region above this.

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For any x, 2x plus 1 will be right on the line, but all the

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y's greater than that are also valid.

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So the solution set of that first equation is all of this

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area up here, all of the area above the line, including the

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line, because it's greater than or equal to.

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So that's the first inequality right there.

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Now let's do the second inequality.

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The second inequality is y is less than 2x minus 5.

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So if we were to graph 2x minus 5, and something already

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might jump out at you that these two are

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parallel to each other.

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They have the same slope.

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So 2x minus 5, the y-intercept is negative 5. x is 0, y is

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negative 1, negative 2, negative 3, negative 4,

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negative 5.

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Slope is 2 again.

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And this is only less than, strictly less than, so we're

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not going to actually include the line.

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The slope is 2, so it will look something like that.

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It has the exact same slope as this other line.

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So I could draw a bit of a dotted line here if you like,

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and we're not going to include the dotted line because we're

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strictly less than.

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So the solution set for this second inequality is going to

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be all of the area below the line.

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For any x, this is 2x minus 5, and we care about the y's that

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are less than that.

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So let me shade that in.

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So before we even get to this last inequality, in order for

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there to be something that satisfies both of these

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inequalities, it has to be in both of their solution sets.

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But as you can see, their solutions sets are completely

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non-overlapping.

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There's no point on the x, y plane that is in both of these

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solution sets.

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They're separated by this kind of no-man's land between these

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two parallel lines.

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So there is actually no solution set.

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It's actually the null set.

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There's the empty set.

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Maybe we could put an empty set like that, two brackets

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with nothing in it.

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There's no solution set or the solution set of

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the system is empty.

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We could do the x is greater than 1.

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This is x is equal to 1, so we put a dotted line there

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because we don't want include that.

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So it would be all of this stuff.

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But once again, there's nothing that satisfies all

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three of these.

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This area right here satisfies the bottom two.

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This area up here satisfies the last one

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and the first one.

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But there's nothing that satisfies both these top two.

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Empty set.

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