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Understanding absolute value is crucial for solving many mathematical problems, especially when dealing with distances and inequalities. The absolute value of a number is a measure of its distance from 0 on the number line, with the key aspect being that this distance is always positive or zero. In mathematical terms, the absolute value of a real number x is denoted as \( |x| \).

For example, the absolute value of both -4 and 4 is 4, despite one being negative and the other positive, because they are the same distance away from 0. An important property to remember is that the absolute value of any real number is always non-negative, which directly informs how we solve absolute value inequalities.

Inequalities are mathematical expressions that compare two values, showing that one is less than, greater than, less than or equal to, or greater than or equal to another. The symbols for these relationships are <, >, \(\leq\), and \(\geq\) respectively.

When dealing with absolute value inequalities, such as \(|x-7| \geq 0\), we're considering the distances on the number line rather than actual values. Solving these inequalities often involves considering separate cases because absolute value equations encapsulate both positive and negative scenarios for the expression inside the absolute value.

A solution set is the collection of all values that satisfy a given inequality or equation. In the context of inequalities, the solution set represents all the possible real numbers that make the inequality true.

For example, the solution set to the inequality \(|x-7| \geq 0\) includes every real number, since the absolute value of any real number minus 7 will always be a non-negative value. Thus, we say that the inequality holds true for all real numbers. Expressing solution sets in interval notation provides a concise way to communicate the range of solutions on the number line.

The set of real numbers includes every number that can be found on the number line, encompassing all rational and irrational numbers. Rational numbers are those that can be expressed as a fraction with integer numerator and non-zero integer denominator, like 2/3 or -5. Irrational numbers, on the other hand, cannot be expressed as such fractions and have non-repeating, non-ending decimal expansions, like \(\pi\) or \(\sqrt{2}\).

The importance of real numbers in solving absolute value inequalities lies in the fact that any real number can be plugged into an absolute value expression, and the resulting value will always adhere to the property that absolute values are non-negative. For the inequality \(|x-7| \geq 0\), since the solution includes all real numbers, it testifies to the statement that the absolute value will never yield a negative result.