Solving of inequalities involving modulus

"Solve the inequality 1/(x-a) < 4|x-a|, where a is a positive constant, leaving your answer in terms of a." The answers are x <a and x > a + 1/2.

What I've got:

I know the above inequality gives 4(x - a) < -1/(x - a) or 4(x - a) > 1/(x - a).

The inequality on the left gives x < a , while the inequality on the right gives a - 1/2 < x < a or x > a + 1/2. Taking the union of the 2 solutions gives x < a or x > a + 1/2.

I can get x < a **but I can't get a - 1/2 < x < a or x > a + 1/2.** Can anyone help me out here? thanks! Please detail the steps, I've been working on it for a long time but can't get the answer