Question

The inequality \(|x|>0\) has as its solution set \(\\{x \mid x \neq 0\\}\). Explain why.

Short answer

The solution set \({x \mid x \eq 0}\) includes all real numbers except 0 because the absolute value \(|x|\) must be positive.

Step-by-step solution

Understand the absolute value function

The absolute value of a number, denoted by \(|x|\), represents the distance of that number from 0 on a number line. Therefore, \(|x|\) is always non-negative.

Interpret the inequality \(|x| > 0\)

The inequality \(|x| > 0\) means that the distance from 0 must be greater than 0. In other words, \(|x|\) is not allowed to be 0; it must be a positive value.

Identify the values of \(x\) that satisfy the condition

The only time \(|x| = 0\) is when \(x = 0\). So for \(|x| > 0\), \(x\) must be any value other than 0. This means the solution set includes all real numbers except 0.

Express the solution set

Thus, the solution set of the inequality \(|x| > 0\) is written as \({x \mid x \eq 0}\). This notation indicates all real numbers except 0.

Key concepts

absolute value

When we talk about absolute value, we're discussing the distance a number is from zero on a number line. This distance is always non-negative. For any real number \(x\), the absolute value is denoted as \(|x|\). If \(x\) is positive or zero, \(|x| = x\). If \(x\) is negative, \(|x| = -x\).

inequalities

Inequalities show the relationship between two values and can be expressed using symbols such as \(>\), \(<\), \(\geq\), or \(\leq\). When we say \(|x| > 0\), we mean that the distance from zero must be greater than zero. This implies that \(|x|\) is always positive. When \(|x| = 0\), it only happens if \(x = 0\). Therefore, for \(|x| > 0\), \(x\) must be any value other than zero.

number line

A number line is a visual representation of numbers laid out in a straight line. Zero is in the center, with positive numbers to the right and negative numbers to the left. When solving absolute value inequalities, the number line helps us visualize solutions. For instance, for the inequality \(|x| > 0\), we identify values on the number line excluding zero. This includes all points to the left of zero (negative numbers) and all points to the right of zero (positive numbers).