If points A, B, and C lie on a coordinate line and points A and B
have coordinates 15 and 7 respectively, then which of the possible
coordinates for point C satisfy(ies) d(A, C) < d(B, C)?
This is the number line:

As Stanbon says in his solution, if C is anywhere between A and the
midpoint of AB, then C will be closer to A than to B and thus the 
distance from A to C will be less than the distance from B to C.

However, you should find this with an absolute value inequality, 
which is what I'm sure you are supposed to do:

Let x represent any coordinate of C

d(A,C) < d(B,C)

|7-x| < |15-x|

Since x is not less than 7, the left side becomes
x-7.  Since x is less than 15, the right side
becomes 15-x.  So we have

x - 7 < 15 - x

   2x < 22

    x < 11   

And we are given that x > 7, so we can
write 

  7 < x < 11

So the possible locations for C are shaded below:


The darkened circle at A, says C's coordinate can be as low as 7,
but no lower.  The open circle at the right end of the shaded line 
says that C's coordinate can be less than 11, but cannot 
be equal to 11.

So the set-builder notation of this is 

C Î {x | 7 < x < 11}

In interval notation this is

C Î [7,11)

Edwin