We are asked to explain how many solutions an absolute value equation can have. Let's think about it. Assume that we have the following absolute value equation where A represents a variableexpression.
|A|=b
What does this absolute value mean? In general, an absolute value represents the distance from 0 on a number line. In our case, the equation means that the distance on a number line from A to 0 is b units. Depending on value of b, we have three possible cases. Let's examine them one by one.
First Case, b < 0: Since the distance on a number line can never be a negative number, we have no solution to this equation.
Second Case, b = 0: In this case, the distance on a number line from A to 0 is 0 units. This is only true when A=0. Therefore, the equation has one solution.
Third Case, b > 0: Since b>0, there can be two different points which have the same distance from 0. Therefore, the equation can have two solutions, A=b and A=- b.
Remember, even if we find two different solutions, we need to check whether these solutions satisfy the absolute value equation. If they do not satisfy the equation, they cannot be in the solution set.
