In general the line through (a,b) and (c,d) is
(c-a)(y-b)=(d-b)(x-a)
(-4 - 3/2)(y - 2) = (-1 - 2)(x - 3/2)
Multiply both sides by 2 to clear the fractions. We use the 2 in the first factor on the left and the second on the right.
(-8 - 3)(y-2)=(-3)(2x - 3)
-11(y - 2) = -3(2x - 3)
-11y + 22 = -6x + 9
6x - 11y = 9 - 22 = -13
Answer: 6x - 11y = -13
Check:
6(1 1/2) - 11(2) = 9 - 22 = -13, good
6(-4) - 11(-1) = -24 + 11 = -13, good