Question

b. explain what the solutions to the inequality |x - 2| < 0 mean based on distance on a number line. what are the solutions?

Explanation:

Step1: Recall the property of absolute value

The absolute value \(|a|\) represents the distance of \(a\) from \(0\) on the number line, and \(|a|\geq0\) for all real numbers \(a\). In the inequality \(|x - 2|\lt0\), since the absolute value of any real - number expression \(x-2\) (where \(x\in R\)) is non - negative.

Step2: Analyze the inequality

The statement \(|x - 2|\lt0\) is asking for values of \(x\) such that the distance between \(x\) and \(2\) on the number line is less than \(0\). But distance (measured by absolute value) cannot be negative.

Answer:

There are no solutions. The absolute value \(|x - 2|\) represents the distance between \(x\) and \(2\) on the number line. Since distance (as measured by absolute value) is always non - negative (\(|a|\geq0\) for any real number \(a\), here \(a=x - 2\)), there is no real number \(x\) for which \(|x - 2|\lt0\).