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# Graphing systems of inequalities

CCSS.Math:

## Video transcript

we're asked to determine the solution set of this system and we actually have three inequalities right here and a good place to start is just to graph the sets the solution sets for each of these inequalities and then see where they overlap and that's the region of the XY coordinate plane that will satisfy all of them so let's first graph well let's just graph y is equal to 2x plus 1 and then 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 will 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 it looks something something like that 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 that 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 above the line including the line because it's greater than or equal to so that's this 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 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 the slope is 2 so it'll look something like that has the exact same slope as this other line slope is 2 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 oh the line right for any X this is 2 X 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 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 solution sets are completely non overlapping there's no point on the XY plane that is in both of these solution sets that are 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 you know two brackets with nothing in it there is no solution set or the solution set of the system is empty I mean we could do the X is greater than 1 this is X is equal to 1 so we put a dotted line there for X's because you don't want to 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 on the first one but nothing there's nothing that satisfies both these top two empty set