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When we encounter algebraic equations, especially those involving radicals, the substitution method can be an invaluable tool for evaluating potential solutions. This method involves replacing the variable with the given number and observing how the equation behaves. It's like swapping out a puzzle piece to see if it fits perfectly.

For example, if we're asked to determine if a certain value is a solution for an equation, we substitute this value for the variable and perform the calculations to see if both sides of the equation remain balanced. In the problem, substituting the value '1' for 'x' allowed us to quickly assess if the equation holds true. Following substitution, the crucial step is to simplify the expressions to see if we end up with a valid statement. This process can be particularly enlightening when working with radical equations because it offers a straightforward path to determine the feasibility of potential solutions.

Simplifying expressions, particularly those containing radicals, is a foundational skill in algebra. Simplification makes complex expressions more manageable and easier to evaluate. In the context of radical equations, this frequently involves combining like terms, performing arithmetic operations within the radical, and sometimes rationalizing the denominator.

To simplify the expression \(3\sqrt{2 - 3}\)) in our example, we calculate the value inside the radical first (\(2 - 3 = -1\)), and then apply the radical to this result. This process helps reveal the nature of the solution. Simplification is not just about making the expression look neater; it is about transforming the expression into a form where the properties of the included numbers, especially the nature of square roots, become clear and consequential.

In mathematics, real numbers encompass all the numbers on the number line, including both positive and negative numbers, and zero. However, when it comes to square roots, not all real numbers are eligible. In the realm of real numbers, the square root of a negative number does not exist, because no real number squared will give a negative result. This concept is critical when solving radical equations.

In our practice problem, both sides of the equation involved square roots of negative numbers: \(\sqrt{-1}\)) and \(\sqrt{-9}\)). These expressions don't have a real number solution, as the square roots of negative numbers are not defined in the set of real numbers. Recognizing this quickly allows us to conclude that the equation cannot hold true for any real number value of 'x', including the proposed solution of 1. This understanding is fundamental when working within the real number system and confronting radicals that contain negative quantities under the radical sign.