To determine which counterexample shows the conjecture "If the product of two numbers is positive, then the two numbers must both be positive" to be false, let's examine each product presented:
1. [tex]\( (-3)(+1) = -3 \)[/tex]
- The product is negative. This does not contradict the conjecture since the product is not positive.
2. [tex]\( (-3)(-1) = +3 \)[/tex]
- The product is positive, but the two numbers are not both positive (both numbers are negative). This counterexample contradicts the conjecture because it shows that a product can be positive even when the numbers are not both positive.
3. [tex]\( (+3)(+1) = +3 \)[/tex]
- The product is positive, and both numbers are positive. This does not contradict the conjecture.
4. [tex]\( (+3)(-1) = -3 \)[/tex]
- The product is negative. This does not contradict the conjecture since the product is not positive.
Thus, the counterexample that shows the conjecture to be false is:
[tex]\[ (-3)(-1) = +3 \][/tex]
This example demonstrates that two numbers can multiply to produce a positive product even when both numbers are not positive.
