Which graph shows the solution set of
X-1/x-3 < 0?

Graph D

Open holes at 1 and 3; shading in between

Explanation:

Let p = x-1 and q = x-3

The problem we want to solve is in the form p/q < 0

Since p/q is negative, there are two cases we have to address

The signs of p and q are opposite one another. Recall this is due to the fact that dividing a positive over a negative (or vice versa) leads to a negative value.

Let's go over Case 1

If p is positive, then

p > 0

x-1 > 0

x > 1

If q is negative, then

q < 0

x-3 < 0

x < 3

Combine x > 1 with x < 3 and we get 1 < x < 3. We'll come back to this later.

Now case 2

If p is negative, then

p < 0

x-1 < 0

x < 1

If q is positive, then,

q > 0

x-3 > 0

x > 3

We get a contradiction.

Both "p is negative" AND "q is positive" must be true at the same time. But if p is negative, then x < 1 which contradicts x > 3 (which happens when q is positive)

Put simply: case 2 is not possible. There's no way to have p be negative at the same time q is positive. This because we're wanting x to both be smaller than 1 AND also be greater than 3 at the same time.

We must rule out case 2 and side with case 1 instead.

The conclusion of case 1 was that 1 < x < 3

The graph of this has open holes at 1 and 3 on the number line. The shading is between the open holes to represent all values between 1 and 3, excluding the endpoints. Graph D indicates these features, so that's why graph D is the answer.

Note: the open holes indicate that x = 1 and x = 3 are not part of the solution set. Another reason x = 3 is restricted is to avoid a division by zero error.