1. Understand acetylation of an amino group

An amino group, $-NH_2$, on acetylation becomes an acetamide group, $-NHCOCH_3$.

This change is: $-NH_2 \longrightarrow -NHCOCH_3$

The net mass added per amino group is the mass of an acetyl group minus one hydrogen: $COCH_3 - H$

But it is easier to use the reagent relation:

Each acetylation replaces one hydrogen of $-NH_2$ by an acetyl group, so the increase in molar mass per amino group is: $M(\mathrm{COCH_3}) - M(H) = 43 - 1 = 42$

Alternatively, $-NH_2 \to -NHCOCH_3$ adds $C_2H_2O$, whose molar mass is: $2\times 12 + 2\times 1 + 16 = 42$

So, each amino group increases molar mass by $42\,\mathrm{g\,mol^{-1}}$.

2. Calculate total increase in molar mass

Initial molar mass of compound $x$: $108\,\mathrm{g\,mol^{-1}}$

Final molar mass after acetylation: $192\,\mathrm{g\,mol^{-1}}$

Increase in molar mass: $192 - 108 = 84\,\mathrm{g\,mol^{-1}}$

3. Find number of amino groups

If one amino group contributes $42\,\mathrm{g\,mol^{-1}}$, then number of amino groups is: $\frac{84}{42} = 2$

4. Final answer

The number of amino groups in compound $(x)$ is: $\boxed{2}$