Hi. I'm Anthony Varela, and this tutorial is about graphing linear inequalities. So we're going to talk about how this is similar to and different from graphing equations. And then we're going to talk about solution regions on the graph, and we're also going to talk about test points to confirm that you've correctly graphed your linear inequality.
So let's go ahead and graph this linear inequality y is greater than 5x minus 3. So the first thing that you want to do, step number one, is to graph this as an equation. So we're going to graph y equals mx plus b, or y equals 5x minus 3.
But there's one thing that's different when we're graphing this as a line. Look at our inequality symbol. And if this is a strict inequality symbol-- meaning we're not including "or equal to"-- we're going to be using dashed lines. And the reason we use dashed lines is to show that the exact value that falls on the line that we're going to draw is not part of the solution.
If our inequality had the non-strict symbols-- so the ones that included "or equal to"-- we would be able to use a solid line, just as we graph equations on a graph. So this inequality symbol tells me, I'm going to use a dashed line to draw y equals 5x minus 3. So here's what y equals 5x minus 3 looks like. We see it has a y-intercept of negative 1, negative 2, negative 3, and it has a slope of 5.
So if I go up 1, 2, 3, 4, 5, and over 1, and that point on this line. 1, 2, 3, 4, 5, over 1, I'm on the line. So this is our dashed line that represents y equals 5x minus 3.
Now notice what this does is it splits up the coordinate plane into two half-planes. We have a section over here and a section over here. Now one side represents the solution to this inequality, x and y values that satisfy this inequality. The other side represents x and y values that don't satisfy the inequality.
So we need to shade in 1/2 of this graph to represent the solution region. So how do we know which side to shade? So step number two is shade a half-plane. And this is what I like to think about-- at least with linear inequalities-- is this really helps me decide which side to shade.
If we have our less than or less than or equal to inequality symbols, we're shading below the line. So that would be less than, less than or equal to means underneath, below. And if our inequality sign is greater than or greater than or equal to, we're going to be shading above.
So here I have greater than. So I'm going to be shading above this line, which would be this area right here. This would then be below the line. This is above the line.
And then if this is hard for you to understand, a nice concrete way to confirm that we've done this right is to use a test point. So step three is to choose a test point. This is going to be a coordinate point that we're going to draw an x and y value from, plug it into our inequality, and see if we get a true or false statement.
So if it's true, that means that that point exists in the solution region, and we can shade that entire side. If our statement is false, it means it's not in the solution region, and we would then shade the other side of our graph. So I always like to choose the origin if at all possible. You don't really want to choose anything on this line, just so that we're 100% clear about what's in the solution region and what's not.
So I like choosing the origin because the origin is 0, 0. 0's are so nice to work with. Adding 0 is easy. Multiplying by 0 is easy. So 0, 0 represents x equals 0 and y equals 0 that we're going to plug into this inequality.
So I have 0 is greater than 5 times 0 minus 3. And I want to know if this is true or false. Well, 5 times 0 is just 0, so we're left with 0 is greater than negative 3. This is true, so we have confirmed that the origin-- this point-- exists within the solution region. And I can shade everything on this side of my dashed line.
So now let's go ahead then, just out of curiosity, pick something on the other side. So let's choose this point. This would be 1, 2. x equals 2, and y equals negative 1. Let's plug those into our inequality.
So I put in negative 1 for y, and I put in 2 for x. So now I want to know if negative 1 is greater than 5 times 2 minus 3. Well, 5 times 2 is 10. And when we take away 3, we get negative 1 is greater than 7, and this is not true. Negative 1 is not greater than 7, so this confirms that this point here does not exist within our solution region, so this entire half-plane is not the solution region, which tells me to shade in this side.
So let's review graphing linear inequalities. Step number one, graph it as if it is a line, y equals mx plus b. But you want to use a dashed line for those strict inequality symbols and solid lines for the non-strict inequality symbols.
Then you're going to shade a half-plane, so you're going to shade what represents the solution region, all x and y values that satisfy your inequality. Now a nice tip would be if your inequality symbols are less than or less than or equal to, you shade below the line. If your inequality symbol is greater than or greater than or equal to, you shade everything above the line. Or you can go ahead and use a test point to decide which half-plane to shade or to confirm the half-plane that you've already shaded.
What you want to do is choose a point that's going to give you an x and y value, plug that into your inequality. If it's a true statement, it's in the solution region, that whole half-plane can be shaded. If it gives you a false statement, that means it's not in the solution region, and you would then shade in the other half-plane. So thanks for watching this tutorial on graphing linear inequalities. Hope to see you next time.