Problem 118 - College Algebra

Question

Explain why \(|x|<-4\) has no solution.

Step-by-Step Solution

Answer

The inequality \(|x|<-4\) has no solution because the absolute value of any real number is always nonnegative i.e., \(|x| \geq 0\). Thus, \(|x|\) can never be less than 0 let alone -4.

Step 1: Understand the Absolute Value Function

The absolute value function, denoted as \(|x|\), gives the distance of \(x\) from zero on the real number line. It is defined as \( |x| = x \) for \( x \geq 0 \), and \( |x| = -x \) for \( x < 0 \). In both cases, the result is a nonnegative number, thus, \( |x| \geq 0 \). This means the absolute value of any number cannot be negative.

Step 2: Apply the Absolute Value Function

The inequality \(|x|<-4\) is asking for all the values of \(x\) that, when the absolute function is applied, result in a value that is less than -4. Due to the properties of the absolute value function, this set is empty since \(|x|\) cannot be negative.

Step 3: Conclude the Solution

Considering the properties of the absolute value function \( |x| \geq 0 \), it's impossible for \( |x| \) to be less than -4, or any negative value for that matter. Thus, the inequality \(|x|<-4\) has no solution. This is because it contradicts the basic definition and properties of the absolute value.