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Algebraic inequalities, similar to equations, express relationships between variables and constants, but with an important twist: they introduce the concept of range and comparison rather than equality. They are fundamental in determining intervals of possible values rather than just a single solution. An inequality can use signs like less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥).

In practice, if we say that 'a is less than b', algebraically, this translates to the inequality a < b. This means that a can be any value lower than b but never equal to or higher than b. This concept is crucial in many areas of mathematics and real-world applications, like defining budget limits, determining safe temperature ranges, or establishing benchmarks in various performance metrics.

Translating inequalities involves converting the verbal or written description of a relationship into an inequality sign and numerical or algebraic expression. This translation process is a crucial skill in algebra, as it allows you to bridge the gap between a word problem and its mathematical representation.

For instance, the verbal phrase 'No more than 100' translates to the algebraic inequality x ≤ 100, where x represents the number in question. Similarly, 'a number increased by 7 is less than 15' translates to the algebraic inequality x + 7 < 15.

Using our earlier textbook exercise where z is less than 8, we precisely translated the verbal phrase into the inequality z < 8. Translating inequalities correctly ensures that one understands both the context of the problem and the resulting implications on the values variables can take.

Variable expressions are mathematical phrases that combine numbers, variables, and operation symbols but do not contain an equality or inequality sign. These expressions become a foundational element when we deal with equations and inequalities, as they represent a value that can change depending on the variables involved.

Take the expression 3x + 5 for example; this expression indicates that we should multiply the variable x by 3 and then add 5 to the result. The value of this expression varies depending on the value of x. When translating verbal phrases into variable expressions, it's essential to accurately represent the intended operation and the proper order of these operations according to the conventional order of operations.

In an inequality context, variable expressions help define the range of values a variable can have. For example, saying 'a number decreased by 2 is greater than 10' leads to the variable expression x - 2 > 10, where x denotes the unknown number we're describing.