If $x \succsim_i x_i^*$ then $p \cdot x_i\ge w_i.$ (MWG 16.C.2)
Hint: Prove by contradiction. That is, suppose $x_i\succsim_ix_i^*$ but $p\cdot x_i&lt;w_i$, show that this would lead to a contradiction of $x_i^*$ being maximal for the feasible set.