now what if we have multiple linear equations or a system of linear equations how can we shade the appropriate region so first let's graph in let's plot the line X is equal to or greater than 1 which is a vertical line it's also a SLE line because it can be equal to NE 1 and we have another one at two this one's going to be a horizontal line and Y is greater than two but not equal to two so it's going to be a dash line now let's focus on this one X is equal to or greater than NE 1 when X is greater than a number it's to the right of that number so we're going to have to shade to the right of the vertical line now let's focus on y let me use a different color Y is less than two so we have to shade below the horizontal line so we need to choose a region where the equation is true for both the answer is basically the region where both colors are shaded so this is the answer the equation is true well it's true for both equations in that region and that region X is greater than 1 and Y is less than 2 so that's that's the answer now let's try another example let's say that Y is greater than 2x - 3 and also that Y is less thanx + 5 go ahead and try this one so the first graph has a y intercept of -3 and it has a slope of two so as we travel one unit to the right we need to go up two units to get the next point and then if we travel one more unit to the right go up to and then repeat the process now we need to graph this using a dash line now let's graph the second one but this time I'm going to graph it in a light gray color so the Y intercept is five and the slope is negative 1 so as we travel one to the right we need to go down one and it's going to be a dash line as well so this is it now which of the four regions should we shade should we shade this region this region this one or that one one what would you say well let's focus on the first equation the one with the red line Y is greater than the function that means we need to shade above that line so the Blue Line exists in these two regions so the answer won't be here and it won't be there now let's focus on the second equation Y is less than the function it's less than x + 5 so it's less than the Gray Line notice that the only region where both equations are true is this region and so that's the region that's going to be shaded so that's the answer let's try this one let's say x is greater than 1 Y is less than or equal to 6 and Y is greater than 3 2x - 3 so let's start with the first graph so X is greater than one so that's going to be a dash line at one now the next one y is less than or equal to 6 so that's a horizontal line at six each Mark represents a value of one and then let's go ahead and graph this function the Y intercept is at -3 actually let me use a different color so you can see it and the slope is 3 over2 the top number is the rise the bottom number is the run so the rise is three the run is two which means we need to rise three units go up three and then run two units to the right that will take us to the point uh 20 0 which is the x intercept and then we need to go up three over two so that will take us to the point 4 three and it's going to be a dash line so we need to find out where we need to shade so first X is greater than one that means we need to shade to the right of this function next we can see that let's use a different color Y is less than six is less than or equal to six so we have to shade below the Gray Line and then Y is greater than this function so we need to shade above the blue line so what region has all three colors notice that it's the region that's enclosed by all three graphs it's basically this region here it has all three colors it's less than y = 6 is to the right of x is greater than one the red line and it's above the blue line so that's where all three shade regions exists so only this answer this region should be shaded so that's the answer now let's say if we have two linear equations in standard form 2x + 3 Y is greater than 6 and 3x - 4 y let's say it's less than = to 12 what I like to do instead of using test points I like to solve for the variable y in the first equation to get y by itself we need to move the 2x to the left side so it's -2x + 6 and then we need to divide everything by 3 so Y is greater than -2 3x and 6 ID 3 is 2 so this is the first equation I'm going to graph now the second one let's move -4 y to this side so it becomes POS 4 Y and let's take the 12 move it to the left side so it's going to become -2 so therefore 3x - 12 is less than or equal to 4 Y and then just divide everything by four so thus we'll have this equation 3x over 4 - 3 is less than or equal to Y which if you reverse the equation it's Y is equal to or greater than 3 4x - 3 and so I'm going to use this form of the equation so let me just erase a few things I'm just going to rewrite the two important equations on top so personally I find it easier to do this to solve for y and then graph it but that's just me though you can use test points if you want to okay I didn't want that to happen so now let's go ahead and begin so let's put the marks first now let's graph the first one the Y intercept is two and the slope is -2 over 3 so the rise is -2 the run is three and that's going to take us to the point 30 and then we need to go down two and then over three so that will take us to the point 6 -2 somewhere in this region and so if we just draw a rough sketch doesn't have to be perfect it's going to look something like that now let's graph the other one the Y intercept is -3 and the slope is 3 over 4 so we need to go up three units and over four so that will give us the x intercept 4 Z and it's greater than or equal to so we're going to connect those two points of a straight line all you need is two points to graph a linear equation so the answer is either in this region this region this one or this one it's one of the four so let's start with the first function the red line Y is greater than function so we need to shade above the red line which means that it's either in this region or this region now let's graph the next one y is greater than the function so it's greater than the gray one so it's no longer this region therefore the answer is this region here the top one the answer is true for both functions in that region