Which of the following is true about 0<|x|-4x<5?

Hello Gmat_mission.

We need to solve the inequality: $$0 <|x|-4x < 5.$$ The algebraic way to solve this inequality is the following:

1. If x<0 then we get: $$0 <-x-4x < 5$$ $$0 <-5x < 5$$ $$0 > x > -1\ \ \ or\ \ \ -1\ <\ x\ <\ 0$$ The solution of this case is -1 < x < 0.

2. If x>=0 then we get: $$0 < x-4x < 5\ $$ $$0 <-3x < 5\ $$ $$0 > x > -\frac{5}{3}\ or\ \ \ -\frac{5}{3} < x < 0\ $$ Since we have supposed that x>=0 and we get that -5/3 < x < 0, this implies that the solution in this case is the empty set.

Therefore, the solution of the original problem is -1 < x < 0.

In other words, the correct answer is the option A.

I hope it helps.