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CCSS.Math:

let's graph ourselves some inequalities so let's say I had the inequality Y is less than or equal to 4x plus 3 and we want to on our XY coordinate plane we want to show all of the X&Y points that satisfy these this condition right here so a good starting point might be to break up this less than or equal to because we know how to graph y is equal to 4x plus 3 so this thing is the same thing as Y could be less than 4x plus 3 or Y could be equal to 4x plus 3 that's what less than or equal means it could be less than or equal and the reason why I did that on this first example problems because we know how to graph that so let's graph that so if we that's my try to draw it a little bit neater than that so that is no that's not good so that is my vertical axis my Y axis this is my x axis right there that is the x axis and then we know the y intercept the y intercept is 3 so the point 0 3 1 2 3 is on the line and we know we have a slope of 4 we know we have a slope of 4 which means if we go 1 in the X direction we're going to go up 4 and the Y so 1 2 3 4 so it's going to be right here and that's enough to draw a line but we could even go back in the X direction if we go 1 back in the X direction we're going to go down 4 1 2 3 4 so that's also going to be a point on the line so my best attempt at drawing this line is going to look something like something this is the hardest part it's going to look something like that that is a line should be straight I think you get the idea that right there is the graph of y is equal to 4x plus 3 so let's think about what it means to be less than so all of these points satisfy this inequality but we have more this is just these points over here what about all of these that are where y is less than 4 X plus 3 so let's think about what this means when let's pick up some values for X when X is equal to 0 what does this say when X is equal to 0 then that means Y is going to be less than 0 plus 3 y is less than 3 when X is equal to negative 1 X is equal to negative 1 what is this telling us 4 times negative 1 is negative 4 plus 3 is negative 1 y would be less than negative 1 when X is equal to 1 what is this telling us 4 times 1 is 4 plus 3 is 7 so Y is going to be less than 7 so let's at least try to plot these so when X is equal to when X is equal to let's plot this one first when X is equal to 0 y is less than 3 so when X is equal to 0 y is less than 3 so it's all of these points here that I'm shading in in green satisfy that right there if I were to look at this one over here when X is negative 1 when X is negative 1 Y is less than negative 1 so Y has to be all of these points down here when X is equal to 1 y is less than 7 when X is equal to 1 Y is less than 7 so it's all of these points down here and in general you take any point you take any point X let's say you take this point X right there if you evaluate for X plus 3 you're going to get the point on the line that is that point that X times 4 plus 3 now the why is that satisfied it could be equal to that point on the line or it could be less than so it's going to go below the line so if you were to do this for all the possible X's you would not get not you would not only get all the points on this line which we've drawn you would get all the points below the line so now we have graphed this inequality it's essentially this line for X plus 3 with all of the area below it shaded now if this was just a less than not less than or equal sign we would not include the actual line and the convention to do that is to actually make the line a dashed line this is the situation if we were dealing this would be the situation if we were doing with just less than four X plus three because in that situation this wouldn't apply and we would just have that so the line itself wouldn't have satisfied it just the area below it let's do one like that so let's say let's say we have y is greater than negative x over two minus six so a good way to start way I like to start these problems is to just graph this equation right here so let me just graph just for fun let me graph y is equal to this is the same thing as negative 1/2 minus 6 so if we were to graph it that is my vertical axis that is my horizontal axis and our y-intercept is negative 6 so 1 2 3 4 5 6 so that's my y intercept and my slope is negative 1/2 well that should be an X there negative 1/2 X negative 1/2 X minus 6 so my slope is negative 1/2 which means when I go to to the right I go down one so if I go to to the right I'm going to go down 1 so two to the right down 1 if I go to to the left I go negative 2 I'm going to go up 1 so negative 2 up 1 so my line is going to look like this my line is going to look like that that's my best attempt at drawing the line so that's the line of y is equal to negative 1/2 X minus 6 now our inequality is not greater than or equal it's just greater than negative x over 2 minus 6 or greater than negative 1/2 X minus 6 so using the same logic as before for any X so if you take any X let's say that's a particular X we want to pick if you evaluate negative have x over two minus six you're going to get that point right there you're going to get the point on the line but the Y's that satisfy this inequality or the Y's greater than that so it's going to be not that point in fact you draw an open circle there because you can't include the point of negative 1/2 X minus 6 but it's going to be all the Y's greater than that it's going to be all the Y's greater than that that'll be true for any X you take this X you evaluate negative 1/2 or negative x over 2 minus 6 you're going to get this point over here but the Y's that satisfied are all the Y's above that all the Y's above that so all of the Y's that satisfy this equation or all of the chord is that satisfy equation there's this entire area above the line and we're not going to include the line so the convention is to make this line into a dashed line and let me draw try my best to turn it into a dashed line I'll just erase sections of the line and hopefully it will look dashed to you so I'm turning that solid line into a dashed line to show that it's just a boundary but it's not included in the coordinates that satisfy our inequality the coordinates satisfy our inequality or all of this yellow stuff is all of this yellow stuff that I'm shading above the line anyway hopefully you found that helpful