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The absolute value of a number refers to its distance from zero on the number line. It is denoted by vertical bars, like this:

• The absolute value of 5 is written as \( |5| \), and it equals 5.
• Similarly, the absolute value of -5 is written as \( |-5| \), and it also equals 5.

The crucial point here is that the absolute value is always non-negative. This means it can never be less than zero. Thus, when solving inequalities involving absolute values, such as \( |x| > 0 \),the solution includes all numbers except zero, because the distance is zero only at the point zero itself. When the inequality is \( |x| < 0 \),it simply has no solution because an absolute value cannot be less than zero. These concepts are essential when working with absolute value inequalities as they help determine the possible values of the variable involved.

Interval notation is a shorthand way of writing a set of numbers. It shows you at a glance which numbers fit within a particular range. Here's a breakdown of how to use it:

• Square brackets \([ ]\) indicate that the end number is included in the set.
• Parentheses \(( )\) indicate that the end number is not included.

For example, \([1, 5]\)means all numbers from 1 to 5, including 1 and 5 themselves. But \((1, 5)\)means all numbers greater than 1 and less than 5, excluding 1 and 5. When we say \((-\infty, 0) \cup (0, \infty)\),we mean all the numbers except zero. It combines two intervals (from negative infinity to zero and from zero to positive infinity) to show that every number except zero is included. Interval notation is a way to communicate sets of numbers clearly and concisely.

Real numbers are the set of numbers you usually think of in daily life. They include various types of numbers:

• Positive and negative numbers.
• Whole numbers (like 1, 2, 3).
• Fractions (like \( \frac{1}{2} \), \( \frac{3}{4} \)).
• Decimal values (like 0.5, 3.14159 ... which could be pi)

In mathematics, real numbers have many special properties. They are continuous, meaning there are no gaps, so between any two real numbers there's always another real number. Understanding real numbers is critical when working with inequalities because the solution could span across all real values. When you encounter inequalities like \(|x| > 0\),the solutions can actually be any real number except a particular few (like zero in this case), which shows the richness and vastness of the real number system.