If we call the numbers a and b, we can write two inequalities describing the conjecture.
See solution.
Practice makes perfect
We want to find a counterexample that proves the given conjecturefalse. If we call the numbers a and b, we can write the conjecture mathematically as two inequalities.
ab&>a
ab&>bIf we solve for b and a in these inequalities, we get two intervals describing a and b.
ab/a&>a/a ⇔ b>1 [1em]
ab/b&> b/b ⇔ a>1
As we can see, both a and b have to be greater than 1 in order for the inequalities to hold true. Therefore, by setting one of the numbers equal to or less than 1, we have found a counterexample.
1( 5)= 5
Since 5 is equal to one of the numbers in our product, we have found a counterexample that proves the conjecture is false.
