The breakeven point is when revenue equals cost or when the profit is zero. P(x) = R(x) - C(x) = 0. If the profit is zero, then R(x) = C(x) . We will use this equation to find the breakeven point-R(x) = C(x) . 3 x = 50 + 2 x. x = 50. So, the hot dog vendor needs to sell at least 50 hot dogs on a given day to break even.
Justinian I, who took power in 527 and would rule until his death in 565, was the first great ruler of the Byzantine Empire.
The first of the western kingdoms Justinian attacked was that of the Vandals in North Africa. King Hilderic, who had maintained good relations with Justinian and the North African Catholic clergy, had been overthrown by his cousin Gelimer in 530.