Why are there two possible answers to an absolute value equation?
I disagree with Anita. I believe that's not the full reason why
There are two possible answers to an absolute value equation because the absolute value function returns the distance of a number from zero on the number line. Since each number has a positive and a negative distance from zero, there are two possible solutions.
An absolute value equation typically has two possible answers because the absolute value of a number is a measure of its distance from zero on the number line, and that distance can be positive or negative.
To understand why there can be two solutions, let's consider the definition of absolute value.
The absolute value of a number 'x' is equal to 'x' if 'x' is greater than or equal to zero. If 'x' is less than zero, then the absolute value of 'x' is equal to the opposite of 'x'. In other words, if 'x' is negative, the absolute value of 'x' is equal to '-x'.
Now, when we have an absolute value equation, such as |x| = a (where 'a' is a positive number), we need to find the values of 'x' that satisfy this equation.
To solve this equation, we need to consider both cases:
1. Positive case: If 'x' is greater than or equal to zero, then |x| = x. So our equation becomes x = a.
2. Negative case: If 'x' is less than zero, then |x| = -x. So our equation becomes -x = a. In this case, we need to solve for 'x' by multiplying both sides of the equation by -1, which gives us x = -a.
Hence, we have two possible solutions: x = a and x = -a.
Therefore, an absolute value equation typically has two possible solutions due to the nature of absolute value, where the magnitude can be expressed as either positive or negative.