Welcome back, everyone. So just in case you were struggling with this problem, I'm here to help. So we're going to graph this inequality,

x 2 + ( y - 1 ) 2 ≤ 9 . Let's get started here with our steps. First, we're going to have to figure out whether you're dealing with a solid or a dashed line, and we do this by looking at the equation or the the inequality symbol. Remember, if there's a bar underneath the symbol, that means it's going to be a solid line. That's exactly what we have here. So, we're going to have a solid line, not dashed, and we're going to have to graph this by switching the inequality symbol with an equal sign. So we're basically just going to have to graph the equality, x 2 + ( y - 1 ) 2 = 9 .

Alright? So what does this look like? It's a circle, and the center is at the coordinates that are inside the parentheses. Here, if we have no numbers inside the parentheses for x, that means it’s 0. So, the center is at coordinate (0, 1). Remember, when we see a minus sign, that indicates the value is actually positive.

The circle has a center at (0, 1). But remember, we actually need the radius of the circle, and that's given to us by the number 9 on the right side. So clearly, we can see that r² equals 9. And so the radius, r, is equal to the square root of 9, which is just equal to 3. We're going to have to go up, down, and to the right and left by 3 units and connect all of those points. Once I do this, it will look something like this.

Remember, we're going to use a solid line for this, but this is basically what our graph is going to look like. Now let’s take a look at the second step here, which is we’re going to have to test a point. Even if you got this far, you may not have noticed where to shade the graph. Do we always go below the circle? Do we go above it? What’s going on here?

Let’s go ahead and test the point (2, 0). So, 2 2 + ( 0 - 1 ) 2 ≤ 9 results in 4 + 1 is less than or equal to 9. That's a true statement, so we’ll shade the point including that point within the graph. But, do we have to graph everything that's below the circle?

Actually, no. Testing this out further will show that only the points within the circle make the inequality true. So, these types of inequalities are tricky because you can’t just shade everything below or above the center of the circle; you actually have to test all these points.

That’s how to graph these types of inequalities with equations of circles. Let me know if you have any questions, and thanks for watching.