Hi, this is Anthony Varela. And in this tutorial, we're going to be writing a linear inequality from a graph. So we're going to be looking at a picture of a graph of a linear inequality and writing that statement of inequality.
So the first thing that we're going to do is write it as an equation, based off the line that we see on the graph. Then we're going to turn it into an inequality by interpreting the shaded region that we see on the graph. And then we'll confirm that we've done everything correctly by using test points.
So let's take a look at this graph. We see a horizontal line. And this is a linear inequality on a graph. And so with horizontal lines, we can write the equation of horizontal lines as y equals a, some number a because all points on a horizontal line share the same y value, no matter what the x value is.
So taking a look at the graph, we can see that this is y equals 2. That's this line right here, y equals 2. Well, to turn this then into an inequality, we're going to keep y and 2. We're just going to replace that equal sign with an inequality symbol.
So now we need to decide what type of inequality symbol we're going to use. Well, the first thing we're going to look at is the line. So if we have a dashed line, that's going to tell me to use our strict inequality symbols-- less than or greater than not, including or equal to.
If our line is a solid line, we're going to be using those non-strict inequality symbols. So less than or equal to or greater than or equal to. So I see a dashed line. So I know that I'm going to be using one of these two symbols.
So my next step then is to decide if this represents less than, or if it represents greater than. So I'm going to be looking, at the shaded region. If we have a region shaded below the line, this represents less than or less than or equal to. And if we see the shaded region is above the line, this represents greater than or greater than or equal to.
So my shaded region tells me we're going to be using, let's see, one of these two symbols. And my line, my dashed line, tells me I'll be using one of these two symbols. So now I know that the inequality symbol I'm using is less than. So this is the graph of y is less than 2.
So now we can use test points to confirm that I've got this right. And so what do I mean by using a test point? Well, pick a test point that exists within your solution region.
And I like to use the origin whenever I can because it's so easy to plug 0 into both x and y. And it's easy to multiply it by 0. It's easy to add or subtract 0. So I always like choosing the origin whenever I can.
So this test point, 0,0, this represents an x value of 0 and a y value of 0 that we can substitute into our inequality and see if we get a true statement. Well, we're dealing with only y here, the variable y. So what I want to check is is this statement true? 0 is less than 2, substituting 0 in for y. And this is a true statement.
So this confirms that the origin exists within my solution region. And so that entire half plane represents the solution region too. We can even choose a point that we believe is not the solution.
So it's not going to be in our shaded region on the other side of the line. So I'm choosing the point 1, 3 So now then I'm going to take a look at this statement-- 3 is less than 2. And I see that that's a false statement. So this confirms that the point 1, 3 is not in the solution region. So I've got my inequality correct. y is less than 2.
Now let's take a look at a vertical line. So the vertical lines can be written with the equation x equals a, where that a value can be whatever. But they all share the same x coordinates, the points on a vertical line no matter where y is.
So right now, looking at the line that I see on the graph, ignoring the shaded region for now, I see this as an equation x equals negative 4. So I'm coming over on the x-axis to negative 4. This is negative 4 for all locations of y.
So now as an inequality, we're going to keep x and negative 4, once again just replacing the equal sign with an inequality symbol. So taking a look at my line, is it dashed, or is it solid? Well, this one's a solid line. So I'm going to be using an inequality symbol that includes or equal to. So now I just need to decide if I'm going to be using less than or equal to or greater than or equal to.
Well, here, we're focusing on this x-axis. And so the shaded region highlights all x values that's greater than negative 4 here. So this could also be thought of as shaded above if we're thinking with the x-axis. If you'd like to think of it in terms of left and right, shaded below would be on the left side of this line. Shaded above would be on the right side of this vertical line.
So now I know then that the inequality symbol I'm going to be using is greater than or equal to because we have a solid line that includes or equal to. And these are all x values then that are greater than negative 4. So let's confirm this by picking out a test point.
Once again, I'm going to use my origin. So now I'd like to then check the statement 0 is greater than or equal to negative 4, which is true. Let's pick a point outside of our solution region and make sure that this is a false statement then that negative 5 is greater than or equal to negative 4. That's false. So I've gotten my inequality correct. x is greater than or equal to negative 4.
All right, so now let's take a look then at a slanted line. So this is neither vertical nor horizontal. And the first thing that I'd like to do is write this out as an equation. So I can see that I have a y-intercept of 5.
So that's going to be y equals mx plus 5. And now I need to figure out the slope. Well, taking a look at the slope here, I noticed that I can go down 1, 2, 3, 4, 5 and over 1, 2, 3 for another point on my line.
So my rise is negative 5. My run is 3. So my slope then is negative 5/3. So the equation of this line is y equals negative 5/3 x plus 5.
But now, I need to turn this into an inequality. And notice that I have the inequality symbol less than. Now I know that it's not going to include or equal to because I have a dashed line. And the solution region highlights everything that fits underneath this line. It's shaded below the line.
So I'm using less than here. So it's confirmed that I've done this by pulling on a test point. Once again, I'm using the origin because it's easy to work with. So then I'm going to plug-in 0 for x and 0 for y and see if I have a true statement.
So 0 times x is just 0. So I have 0 is less than 5, and that is a true statement. Let's go ahead and pick a point then that is on the other side of the line. So this is not part of the solution region. So I'm going to plug in 5 for x and plug in 5 for y.
So let's see, 5 times negative 5 is negative 25. I can divide that by 3. So that's negative 8 and 1/3. And when I add 5, that is negative 3 and 1/3. And I want to know then is 5 less than negative 3 and 1/3.
And no, it's not. So that confirms that this points and all other points on this side of the dashed line are not part of the solution region. So I've gotten my inequality correctly written as y is less than negative 5/3 x plus 5.
So let's review writing a linear inequality from a graph. Well, if you see a dashed line, you're going to use either the symbol less than or greater than. So you're not going to include or equal to. If you see a solid line, you are going to include those or equal to symbol.
So it's going to be less than or equal to or greater than or equal to. If the solution region is shaded below the line, you're going to be using the less than or less than or equal to symbols. If it's shaded above the line, you're going to be using greater than or greater than or equal to. And you can always choose a test point to make sure that you've got it written correctly.
Thanks for watching this tutorial on writing in linear inequality from a graph. Hope to see you next time.