Let's take z<0:
|7z|=-7z; |9-2z|=9-2z so we have 9-2z>-7z. We need to remember that z is negative. 9>-5z or 1.8>-z or -1.8<z. This implies -1.8<z<0.
Let's take z≥0:
|7z|=7z; |9-2z|=9-2z if 9-2z≥0, that is, z≤4.5. So we have to solve 9-2z>7z, giving us 9z<9, z<1; this means that 0≤z<1. And |9-2z|=2z-9 if 9-2z<0, that is, z>4.5. We have to solve 2z-9>7z, 5z<-9, z<-1.8, which conflicts with the premise z≥0, leaving us with 0≤z≤1.
So we have for z: -1.8<z<0, 0≤z<1. Combine these: -1.8<z<1.
Let z=-0.5 in the original inequality: |9-2z|>|7z| becomes |10|>|-3.5| or 10>3.5 which is true, so the inequality holds. Let z=0.5: |9-1|>7*0.5 is 8>3.5 which is true.
Now let's put z=1: |9-2|>7 is not true which is expected because z has to be less than 1.
Now let z=-2: 13>14 which is false, expected because z<-1.8.
Finally, let z=-1.8: 12.6>12.6 which is false as expected, because z=-1.8 is excluded.
These tests confirm -1.8<z<1.