### Video Transcript

The graph shows ๐ฆ equals ๐ divided by ๐ฅ minus three minus two. We can see that the intersection of its asymptotes is at three, negative two and the point 0.5, negative 1.5 and 1.5, negative one are below and above the graph, respectively. Determine the interval in which ๐ lies.

We have two asymptotes: one at ๐ฅ equals three and the other at ๐ฆ equals negative two. The asymptote ๐ฅ equals three is vertical and thatโs because if you take a look at the denominator, it can not equal zero. So the value that would make it equal zero would be three, so we can not have any points on ๐ฅ equals three.

Now, the other is from the transformation of this minus two at the end of our equation; it shifts the entire graph down two. So instead of having the normal horizontal asymptote at ๐ฆ equals zero, it will shift it down to ๐ฆ equals negative two. So in order to figure out what ๐ is actually equal to, we have some points that are above and below our graph. So if we know that a point on the graph is in between that, we can use inequalities to figure out what ๐ would be equal to.

Specifically, weโre looking at this little pink section. So thinking about what ๐ฅ-values there could be, the ๐ฅ-value below the graph is 0.5 and the ๐ฅ-value above the graph is 1.5. So a value in between that, that would be easy to use, it would be one. So taking our equation, letโs plug in one for ๐ฅ. So now we have one minus three on the bottom, which is equal to negative two.

Now, as for the ๐ฆ-values, itโs between negative one and a half and negative one. So if we are determining an interval, we can create an interval using these values. So we want our equation, our graph- weโre looking at the points between negative one and a half and negative one, so greater than negative one and a half, but less than negative one.

So to solve, letโs first add two to both sides. Now, we need to multiply the sides by negative two. However, when you multiply by a negative, weโre gonna have to flip our sign. So ๐ would be less than negative one, but greater than negative two. And a more common way to write that would be negative two is less than ๐ is less than negative one.