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When it comes to solving inequalities, it's key to understand that unlike equations, inequalities express a range of possible solutions. Starting with an inequality like 3x - 5 < 2x + 7, the goal is to isolate the variable on one side.

By transferring 2x to the opposite side, you're essentially moving part of the indefinite range of solutions step by step towards clarity. Once the variable x stands alone, as in x < 12, you visualize the range of numbers that satisfy the condition. Every number less than 12, but not including 12 itself, makes the inequality true.

Understanding this concept paves the way for properly expressing these solutions, specifically using interval notation that encapsulates the entire range succinctly.

The building blocks of algebra are termed as algebraic expressions. They comprise numbers, variables, and arithmetic operations. When faced with 3x - 5 < 2x + 7, recognize that both sides of the inequality are individual algebraic expressions.

Simplifying these expressions by combining like terms and executing operations correctly is crucial. For instance, subtracting 2x from both sides is an operation applied across the inequality to modify and simplify the expressions while maintaining their balance.

Many students often miss simplifying fully or lose track of signs during this process, so careful attention to detail is required to ensure accurate simplification.

In dealing with linear inequalities, the principles of solving equations mostly apply, but with a key difference regarding the inequality symbols. A linear inequality is similar to a linear equation but instead of an equals sign, it contains one of the symbols: <, ≤, >, or ≥.

When solving x < 12, it's not just about finding a single solution; it's determining all the values that x can take. This is done without altering the direction of the inequality unless you multiply or divide by a negative number.

Ensuring that students avoid the common pitfall of treating inequalities as strict equations is vital. Remembering to express the final solution in an appropriate format, such as interval notation, is also essential to comprehensively communicating the range of possible solutions.