Graphing two-variable inequalities

we're asked to graph the inequality Y is less than 3x plus five so if you give us any X and let me label the axes here so this is the x axis this is the y axis so this is saying you give me an X so let's say we take X is equal to 1 right there 3 times 1 plus 5 so 3 times X plus 5 so 3 times 1 is 3 plus 5 is 8 so 1 2 3 4 5 6 7 8 this is saying that Y will be less than 8 Y will be less than 3 times 1 plus 5 so the Y values that satisfy this constraint for that X are going to be all of these values down here let me do it in a lighter color it'll be all of these values for X is equal to 1 it'll be all the values down here and it would not include Y is equal to 8 Y has to be less than 8 now if we kept doing that we would essentially just graph the line of y is equal to 3x plus 5 but we wouldn't include it we would just include everything below it just like we did right here so we know how to graph just y is equal to 3x plus 5 let me write over here so if I were to write Y is equal to 3x plus 5 we'd say okay 3 is the slope slope is equal to 3 and then 5 is the y-intercept now I could just graph the line but because that won't be included in the Y's that satisfy this constraint I'm going to graph it as a dotted line so we'll start with the y-intercept of 5 so 1 2 3 4 5 that's the y-intercept and the slope is 3 so if you go over the 1 you go up 3 so let me do that in that darker purple color so it'll look like this it will look like that and you see that point would be on at that point would be on it if you go back you're going to go down by 3 so that point will be on it that point and that point and I'll just connect the dots with a dotted line so it will look so that is that dotted line is the graph of y is equal to 3x plus 5 but we're not going include it so that's why I made it a dotted line because we want all of the Y's that are less than that so for any X so you pick an X let's say X is equal to negative 1 if you evaluate 3x plus 5 for that X should get here but we only care about the Y's that are strictly less than that so you don't include the line it's everything below it so for any X you pick it's going to be below that line you go you take the X go up to that line and everything below it so for all of the X's it's going to be this entire area it's going to be this entire area we draw it a little bit neater than that it's going to be this entire area that's under the line I'll do it in this orange it's a little easier to see so this entire area under the line is y is less than Y is less than 3x plus 5