Let me give it a shot.

given statement is |m/5| > 1.
Now suppose m/5 is > 0. Then in that case the above inequality is m/5>1 which boils down to m>5. This result suffices with our assumption of m/5 is +ve.
Now suppose m/5 <0 that means m is -ve. Then in that case the inequality becomes -m/5 > 1 which can be simplifed further as -m > 5 or m < -5. This also ok as we have assumed that m is -ve.

Now try to plug the answer choices with the above results.

The question is asking which must be true.
1) is true only when m > 0. but it fails when m <0.
2) is not sufficing any of the conditions.
3) same as 2.
4) This is true in each case . when m > 0 or m<0. So this is the correct option.
5) is true only when m <0. but it fails when m >0.

I hope my explaination is ok.

Sid.