Problem 2 - Beginning and Intermediate Algebra

Question

Does \(|x|=-8\) have a solution? Why or why not?

Step-by-Step Solution

Answer

The equation \(|x| = -8\) does not have a solution because the absolute value function always outputs a non-negative result, and -8 is a negative number. There is no possible value of x that could satisfy this equation.

Step 1: Understanding Absolute Value

The absolute value of a number is a non-negative value that represents the distance a given number is from zero. In simple terms, the absolute value converts a negative number to a positive number, while a positive number stays the same. The notation for the absolute value of a number x is \(|x|\). For example: \(|(-5)| = 5\) \(|(3)| = 3\) From these examples, it is clear that the absolute value function always outputs a non-negative result.

Step 2: Analyzing the Equation

Now let's analyze the given equation: \(|x| = -8\) On the left side, we have the absolute value function applied to the variable x. As we mentioned earlier, the absolute value function always outputs a non-negative result. However, the right side of the equation is -8 which is a negative number.

Step 3: Determining if the Equation has a Solution

As we know that absolute value can only result in non-negative values, we can conclude that there is no possible value of x that could satisfy the equation \(\(|x| = -8\)\), since -8 is a negative number. The fact that the result of the absolute value function can never be negative makes it impossible for this equation to have a solution. Therefore, the equation \(|x| = -8\) does not have a solution.