Graphing two-variable inequalities (old)

graph the inequality Y minus 4x is less than negative 3 so the first thing we could do is we could kind of put this in MX plus B form or slope-intercept form but as an inequality so let's start with we're starting with y minus 4x is less than negative 3 we can add 4x to both sides of this inequality so let's add 4x to both sides of this inequality and then we'll just have a Y on the left hand side these guys cancel out see if Y is less than 4x minus 3 we could have negative 3 plus 4x but we want to write the 4x first just because that's a form that we're more familiar with so it's less than 4x minus 3 and now we can we can attempt to graph it but before I graph it I want to do something a little bit gonna be a little bit careful here so let me draw our axes so this is the x-axis and this is that is the y-axis and we want to be careful because this says y is less than 4x minus 3 not less than or equal to 4x minus 3 or not Y is equal to 4x minus 3 so what we want to do is kind of create a boundary at Y is equal to 4x minus 3 and the inequality the solution to this inequality will be all of the area below that all the Y values less than that so let's try to do it so the boundary line the boundary line would look like let me write over here so we have a boundary we have a boundary at Y is equal to 4x minus 3 notice this isn't part of the solution this isn't less than or equal it's just less than but this will at least help us draw essentially the boundary so we can do it two ways if you know slope and y-intercept you know that 4 is our slope and that negative 3 is our y-intercept or you can literally just take two points and that'll help you define a line here so you could say well when X you could say when X is equal to 0 what is y you get 4 times 0 minus 3 you get Y is equal to negative 3 and we knew that because it was the y intercept so you have 0 and then you have 1 2 zero and negative three and then you have the point and then you have the point let's say when X is equal to I don't know let's say when X is equal to 2 when X is equal to 2 what is y we have 4 times 2 is 8 minus 3y is then going to be equal to 5 so you go 1 2 then you go 1 2 3 4 5 and so you have that point there as well and then we can just and then we can just connect the dots or you could say look there's a slope of 4 so every time we move over one every time we move it 1 in the x-direction we move up 4 in the y-direction so you could draw it like that but so the line will look something like this and I'm just gonna draw it a dotted line because remember this isn't this isn't part of the solution actually let me draw it a little bit neater cuz that point should be right about there this point should be right about there and then this boundary line I'm going to draw as a dotted line so it's going to look like it's going to look something something like that I draw it a dotted line to show that it's not part of the solution our solution is the Y is less than that so for any X so you put pick an X here if you took 4 X minus 3 you're going to end up on the line but we don't want the Y's that are equal to that line we want for that particular X the Y's that are less than the line so it's going to be all of this area over here where less than the line and we're not including the line and that's why I put a dotted line here you can also try values out you can say well there's two this line is dividing our coordinate axes into essentially the region above it and the region below it and you can test it out let's take something that's above it let's take the point 0 0 let's take the point 0 0 and see if that satisfies our inequality if we have Y is 0 is less than 0-3 or we'd get 0 is less than negative 3 this is definitely not the case this is not true this is not true and it makes sense because that 0 that 0 0 is not part of the solution now we could go on the other side on the other side of our boundary and we could take the point I don't know let's take the point 3 comma 0 so let's say that this is the point well that that's right there's a point 2 comma 0 let's take the point 3 comma 0 right over here this should work because it's in the region less than but let's verify it for ourselves so we have Y y is 0 0 is less than 4 times 3 minus 3 0 is less than 12 minus 3 0 is less than 9 which is definitely true so that point does satisfy the inequality so in general you want to kind of look at this as an equal to draw the boundary line we did it but we drew it as a dotted line because we don't want to include it because this is it less than or equal to it's just less than and then our solution to the inequality will be the region below it all the Y's less than the line 4x minus 3