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Absolute value is a fundamental concept in mathematics, especially when dealing with inequalities. It measures the distance a number is from zero on the number line. This distance is always positive or zero.

The symbol for absolute value is vertical bars around a number or expression, like \(|x|\). The absolute value of a number is simple to determine:
- If the number is positive or zero, the absolute value is the same as the number itself, i.e., \(|x| = x\) for \(x \geq 0\).
- If the number is negative, the absolute value is the positive version of that number, i.e., \(|x| = -x\) for \(x < 0\).

In problems involving absolute values, expressions like \(||a|-|b||\) appear. It represents the non-negative difference between the absolute values of \(a\) and \(b\). Understanding how to manipulate and interpret these values is crucial when solving inequalities.

Real numbers are a set of numbers that include both rational and irrational numbers. This set encompasses all possible numbers on the number line. Some key properties of real numbers include:
- **Rational Numbers:** Numbers that can be expressed as fractions, such as 1/2, 4, and 0.75.
- **Irrational Numbers:** Numbers that cannot be written as simple fractions, like \(\sqrt{2}\) and \(\pi\).
- **Whole Numbers and Integers:** Whole numbers are non-negative integers, including zero, while integers include negative numbers, zero, and positive numbers.

All these subsets of real numbers follow basic arithmetic operations and properties, such as the commutative, associative, and distributive laws. Real numbers are used extensively in geometry, calculus, and more, always respecting rules like the triangle inequality, which is a cornerstone of dealing with these numbers in inequalities.

Inequality proofs often involve showing that one expression is always greater than, less than, or equal to another expression for all variables involved. One crucial tool in proving inequalities is the **Triangle Inequality Theorem**. This theorem states that for any real numbers \(x\) and \(y\), \(|x + y| \leq |x| + |y|\).

When tackling inequality proofs, follow these steps:
- **Identify Known Inequalities:** Begin by finding known inequalities, such as the triangle inequality, to relate given expressions.
- **Rearrange and Simplify Expressions:** Use algebraic techniques to rearrange and simplify the inequality.
- **Substituting Expressions:** At times, substituting parts of an inequality with simpler expressions can reveal a known form or direct application of a theorem.

For example, in the inequality \(||a|-|b|| \leq |a-b|\), the problem uses the absolute value property and the triangle inequality. Understanding and applying these principles is crucial in simplifying terms and reaching a valid proof.