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Algebra is the branch of mathematics that involves symbols, usually letters, to represent numbers. These symbols are used in formulas and equations to solve for unknown values. In the given inequality, 7m + 2 < 12, the letter 'm' serves as a placeholder for an unknown number which we ultimately solved.

Algebra allows us to structure problems logically, which is very useful when dealing with complex equations or inequalities. The key to algebra is understanding how to manipulate these symbols following basic rules of arithmetic. For example, we can add, subtract, multiply or divide both sides of an equation without changing its value, as long as these operations are applied evenly to both sides.

When you substitute a number into an algebraic expression like we did with m = 1 in 7m + 2, it helps to transform the algebraic formula into a numerical expression that can be easily calculated. This makes the process much more manageable, as seen when simplifying 7(1) + 2 to obtain 9.

Mathematical reasoning is the process of using logical steps to arrive at a conclusion or solve a problem. This is the backbone of solving any math problem, including inequalities. In our example, we applied mathematical reasoning to confirm if 9 < 12 is true after substituting the value for m.

In practice, reasoning involves several steps such as identifying the given equation or inequality, substituting known values, and analyzing whether the resulting calculation holds true under the given conditions. Each step must follow logically from the previous one. This requires attention to detail, especially when you perform operations like rearranging terms or simplifying expressions.

By applying these logical sequences, you ensure each step is valid and leads you to the final answer. As seen in our example, recognizing the validity of 9 being less than 12 verified that 1 is indeed a solution for the inequality.

The substitution method is a technique used in algebra to replace variables with actual numbers to simplify equations and inequalities. It's a straightforward yet powerful method commonly used to verify solutions or solve for unknowns.

In our specific problem, the substitution method was employed by replacing the variable m with the number 1. This transformed the inequality 7m + 2 into the simple mathematical statement 7(1) + 2 < 12. Carrying out the operations gives us 9 < 12, validating our solution.

The substitution method is not only useful for basic algebra but is also applied in solving systems of equations. It can simplify multiple equations simultaneously by substituting common variables with numbers or expressions. This technique enhances efficiency and clarity in solving complex algebraic problems.