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To understand linear inequalities, let's start with the basics of inequalities which are comparisons between two expressions that may or may not be equal. A linear inequality, such as \( x \geq 12 \), involves a linear expression (an algebraic expression where each term is either a constant or a product of a constant and a single variable to the first power) and an inequality symbol. There are several types of inequality symbols: \( < \) (less than), \( > \) (greater than), \( \leq \) (less than or equal to), and \( \geq \) (greater than or equal to).

In the context of our exercise, the inequality \( x \geq 12 \) defines a set of numbers where \( x \) could be 12 or any number greater than 12. This is shown on a number line with a closed dot at 12 and a line extending to the right, indicating all the numbers greater than or equal to 12. Understanding how to read and write these inequalities is crucial for solving and graphing them.

Translating verbal statements into inequalities is a key skill in algebra. It involves converting everyday language into mathematical expressions that include inequality symbols. The process usually starts by identifying action words or phrases that suggest a particular inequality symbol. For example, phrases like 'at least', 'no less than', or 'a minimum of' point towards a \( \geq \) symbol, indicating that the quantity is greater than or equal to some number.

Following this approach, the phrase 'x is a minimum of 12' immediately signals the use of \( x \geq 12 \). Knowing this translation is just the beginning, as it's equally important to understand the relationship the inequality describes. It expresses a range of possible values for \( x \) rather than a single solution, which is a fundamental distinction from equations.

While inequalities involve relationships between expressions, the building blocks of these relationships are the expressions themselves. Algebraic expressions are combinations of numbers, variables (letters used to represent unknown or varying numbers), and arithmetic operations (such as addition, subtraction, multiplication, and division).

For instance, in the inequality \( x \geq 12 \), we are looking at a very simple algebraic expression: \( x \). This expression represents a variable quantity that could take on any value greater than or equal to 12. Algebraic expressions can become much more complex, with multiple terms and variables, but the principle remains the same: they are the way by which we represent mathematical ideas symbolically. When such expressions are paired with inequality symbols, they form the foundation of linear inequalities.