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Absolute value inequalities involve expressions within absolute value signs being compared to a number using inequality symbols such as <, >, \(leq\), or \(geq\). Understanding these inequalities is essential because they can determine a range of possible values for variables in an equation.

Consider the example \(\vert x-5 \geq 0\vert\). The absolute value of any real number, including \(x-5\), is non-negative, meaning it's either positive or zero. Therefore, the inequality tells us that \(x-5\) is at least 0 units away from 0 on the number line. As a result, every real number meets this criterion because the distance from 5 is always non-negative.

This implies that our solution is the entire set of real numbers, which can be challenging for students at first glance. In solving these types of problems, it's crucial to consider the definition of absolute value and the real number line to identify the interval of the solution.

The absolute value represents the distance of a number from zero on the number line, and as such, it has distinct properties that are fundamental in solving equations and inequalities. One crucial property is that the absolute value of a number is always non-negative.

This means that for any real number \(a\), the absolute value \(\vert a \geq 0\vert\) holds true. Another key property is that \(\vert a \geq b\vert\) if and only if \(a \geq b\) or \(a \leq -b\). This duality is particularly useful when solving absolute value inequalities, as it allows us to split the inequality into two separate conditions to consider.

Understanding these properties helps to clarify why the inequality \(\vert x-5 \geq 0\vert\) has a solution set that includes all real numbers. Since the absolute value measures distances, it's never negative, and in our example, the distance of \(x\) from 5 cannot be a negative number.

The real number line is a vital concept in mathematics, representing all possible values that a real number could take. This visualization helps in solving inequalities, as it provides a clear way to see the set of numbers that satisfy the inequality.

For our illustration \(\vert x-5 \geq 0\vert\), the real number line shows every point as a potential value of \(x\), because no matter where you pick a number on the line, the distance from 5 is not negative. When we express solutions, we often do so using interval notation, which succinctly represents ranges of numbers.

The entire real number line is written in interval notation as \(\left( -\infty, +\infty \right)\), revealing that there's no boundary to the set of real numbers. Such notation is incredibly efficient and beneficial for students to express infinite intervals or ranges of solutions to inequalities.