Assume that neither of the games cost more than $40.
See solution.
Practice makes perfect
We are asked to use indirect reasoning to prove the following statement.
If two computer games cost more than$80,
then at least one of the games cost more than$40.
Let's use p_1 and p_2 for the price of the games.
With this notation we can write the claim we need to prove more formally.
If p_1+p_2>80,
then eitherp_1>40orp_2>40.
Step 1
To start an indirect proof, we assume that the conclusion of the statement is not true.
Note that if it is not true that at least one of the games cost more than $40, then the price of both was at most $40.
Assumption: p_1≤ 40 andp_2≤ 40
Step 2
Let's investigate the consequence of our assumption.
Since the hypothesis involves the total price of the games, let's add the two inequalities in our assumption.
p_1≤ 40 p_2≤ 40
âźą
p_1+p_2≤ 40+40=80
This however contradicts the hypothesis, that states that p_1+p_2>80.
Step 3
Since our assumption that the price of both games was at most $40 contradicts that the total price was over $80, this assumption must be false.
This proves that if Kwan-Yong bought two computer games for over $80, then at least one of the games costs more than $40.
