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The absolute value of a number is a very fundamental concept in mathematics, often depicted with vertical bars on either side of a number, like |x|. It represents the number's distance from zero on a number line, regardless of its direction. This means the absolute value is non-negative—negative numbers become positive, and positive numbers remain the same. For example, both |-2| and |2| equal 2 because each is two units away from zero.

Formally, for a given number a, the absolute value is defined as follows: if a is greater than or equal to zero (\( a \)geq 0)), then \(|a| = a\). On the other hand, if a is less than zero (\( a < 0 \)geq 0)), then \(|a| = -a\). This definition is instrumental in a variety of mathematical operations and can frequently appear in both pure and applied mathematics.

When it comes to evaluating expressions, it involves carrying out the operations indicated within the expression, step by step, according to rules of mathematical operations known as the order of operations. With expressions that include absolute values, a good first step is to start by determining the absolute value of any numbers or subexpressions enclosed by vertical bars.

After understanding the absolute value, we can continue with any additional mathematical operations required by the expression, such as addition, subtraction, multiplication, or division. To evaluate the expression correctly, it's essential to know if any variables are present and if they are, ensure that their values are substituted accurately into the expression before execution.

In relation to the above concepts, mathematical operations are procedures that take one or more input values and produce a new output value. Basic operations include addition, subtraction, multiplication, and division. These become more complex when we deal with algebraic expressions, as they may also involve exponents, roots, and other functions.

Understanding how to correctly perform these operations is key to evaluating expressions. Applying operations to numbers within an absolute value must be done with careful consideration to the absolute value definition, which sometimes can affect the sign or magnitude of the result. In the context of absolute values, subtraction and any operation that could result in a negative number will require special attention to ensure correct evaluation.