Comparing baskets of goods in this exercise Max buys two goods, apple and ketchup. We do not observe his preference but we know it satisfies the three assumptions, completeness, transitivity, and more is better. And as usual, we know his preference does not change with prices or income. Furthermore, we know Max seeks to maximize his utility. Initially, Max’s income is \$40, the price of apple is \$4 and the price of ketchup is \$4, and Max chooses 3 units of apples and 7 units of ketchup to maximize utility. Let A denote this basket. Consider basket B that contains 5 units of apples and 3 units of ketchup. Suppose the price of ketchup becomes \$1.5 and Max’s income becomes \$24.5, while the price of apple remains at \$4. Given the new price and income, is it possible for basket B to be the optimal choice of Max? Briefly explain.


For this question, I drew out the initial and new budget lines. I introduce a new point C (5 units of apples and 5 units of ketchup). Can I say that B cannot be the optimal choice because if we assume by contradiction that B is indeed the optimal choice, then B>C although C is cheaper. However, since the assumption more is better is satisfied, C>B which is a contradiction. Thus, B cannot be an optimal choice. Is my reasoning incorrect?
So we are told that bundle A (3,7) is optimal at prices p=(4,4) with wealth w=40

Consider bundle B (5,3) at prices (4,1.5) with wealth 24.5

By Weak Axiom of Revealed Preference (WARP), we know that if both bundles are affordable under both sets of prices and wealth, and we chose A when B was available, then A is revealed preferred to B.

Both A and B are affordable under those price wealth pairs, so B cannot be optimal. 

What you did graphically is equivalent to the math.