A spring balance measures the weight of an empty balloon to be W_1...

Let the true weight of the empty balloon material be $W_0$ and its volume be $v$.

The buoyant force acting on the empty balloon due to the surrounding air is $F_{b1} = \rho_{air} v g$.

The spring balance measures the apparent weight. Thus, the apparent weight of the empty balloon is:
$W_1 = W_0 - \rho_{air} v g$

When the balloon is inflated, let the volume of the air inside be $V$. The true weight of the air inside is given as $w$, so $w = \rho_{air} V g$ (since the densities of air inside and outside are the same).

The total true weight of the inflated balloon is $W_0 + w$.

The new buoyant force acting on the inflated balloon is due to the total volume displaced, which is $v + V$.
$F_{b2} = \rho_{air} (v + V) g = \rho_{air} v g + \rho_{air} V g = F_{b1} + w$

The apparent weight of the inflated balloon is:
$W_2 = (W_0 + w) - F_{b2}$

Substituting $F_{b2}$ into the equation:
$W_2 = W_0 + w - (F_{b1} + w)$
$W_2 = W_0 - F_{b1}$

Since $W_0 - F_{b1} = W_1$, we get:
$W_2 = W_1$

Since the weight of the air $w > 0$, it strictly follows that:
$W_2 < W_1 + w$

Thus, both $W_2 = W_1$ and $W_2 < W_1 + w$ are correct statements.