We have been told that a, b, and c are real numbers, a ≠0, and b>c. We are asked to decide whether the statement ab>ac is always, sometimes, or never true. For this statement we have two cases.
Case I: a is greater than 0.
Case II: a is smaller than 0.
We will begin with a is greater than 0.
Case I: a>0
Let's check whether the statement is true if a>0. To do this, let's first try with arbitrarily chosen values for a, b, and c, such as a= 2, b= 4, and c= 3.
As we can see, the statement is true for a>0. What about the second case?
Case II: a<0
What happens if a<0? Remember that if we multiply an inequality by a negative number we need to reverse the inequality symbol by the Multiplication Property of Inequality. Once again, let's try with arbitrarily chosen values for a, b, and c, such as a= -2, b= 4, and c= 3.
This result tells us that for a negative value of a the opposite of what we want to be true is true.
ab > ac *
ab < ac ✓
Thus, when a<0, the statement is false. As a result, we say that the original statement is sometimes true.
