This question follows the rules of the general form:
|x-y|=|x| - |y|
if:
a. y = 0; or
b. x and y have the same sign, and x>y
So reorganising the stem.
is |a-c| + |a| =|c|
is |a-c| =|c| - |a|
reshuffle by multiply by -1
is -|a-c| = |a| - |c|
For this to be true c must equal 0 or a and c must have the same sign and a must be greater than c
(1) ab > bc
++ >+(-) a and c could have different signs; or--> No
++>++ a and c could have the same signs; or -->Yes
++>+(0) c could be equal to zero -->Yes
Insufficient
(2) ab<0
could be +- or -+
either way we knowing about c
Insufficient
Combined (1+2)
ab<0
therefore
ab > bc
(-) > bc
Possible cases:
ab > bc
(-)+> +(-) a and c are both negative, but a must be greater than c to satisfy the constraint
+(-)>(-)+ a and c are both positive, but a must be greater than c to satisfy the constraint
In either case, a must be greater than c and must have the same sign as c, therefore combined statements are sufficient.
We can test numbers to validate the theory.
-|a-c| = |a| - |c|
condition 1: a and c are both negative and a > c
-|-2-(-3)| = |-2|-|-3|
-|1| = 2-3
-1 = -1
condition 2: a and c are both positive and a > c
-|3-2|=|3|-|2|
-|1| = |3|-|2|
-1 = 1 is not possible, therefore a and c must both be negative
Bunuel, sorry to pester but is my working out correct?