The expression |c - d| always means "the distance between c and d on the number line".
So the equation |a - 5| = 11 means "the distance between a and 5 is equal to 11". So a can be in either of these two places, 11 away from 5:
-a----------5------------a-----
and either a = -6 or a = 16.
Similarly, if we have (factoring out a 2 from the absolute value and canceling it) |b - 3| = 4, we know that the distance between b and 3 is equal to 4:
-----b-----3-----b------
and b = -1 or b = 7.
You could also use the 'cases method' here to find the possible values of a and b.
We can make a/b negative by using one negative and one positive value, and since we want the minimum value of a/b, we'll want a/b to be negative. So we can consider the two pairs of values that will make a/b negative (one positive value, one negative value) :
- if a = -6 and b = 7, then a/b = -6/7
- if a = 16 and b = -1, then a/b = 16/(-1) = -16
The smaller of these is -16, so that's the minimum value of a/b.
_________________
GMAT Tutor in Toronto
If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com