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Rational expressions are fractions that contain polynomials in the numerator, the denominator, or both. To solve a rational expression means to find the values for which the expression is defined or undefined.

One fundamental aspect is the identification of key numbers, also known as critical values, which could make the denominator of a rational expression zero. Since division by zero is not allowed in mathematics, these particular values are those that are excluded from the domain of the expression. Consider the expression \(\frac{1}{x-5} + 1\). The key number here is found by setting the denominator equal to zero and solving for x, leading us to x = 5. It's crucial to recognize and exclude this value when solving or graphing the expression, as it is a point where the expression is undefined.

In mathematics, there are operations which are considered undefined. One such operation is division by zero. This is because division is the process of determining how many times one number is contained within another, and zero does not contain any value, thus causing a logical inconsistency.

When working with rational expressions, we must always be vigilant to avoid undefined operations. In our original expression \(\frac{1}{x-5} + 1\), the value x = 5 creates a scenario where you would have to divide by zero, which is why it is deemed undefined. Understanding undefined operations help us determine the conditions in which a certain mathematical expression is valid, and this concept is fundamental to higher mathematical concepts like limits and continuity in calculus.

Another important concept when working with rational expressions is setting denominators to zero. The denominator of a fraction indicates the number of parts that make up a whole, and when it is zero, it implies that you're dividing by nothing, which makes no sense in terms of quantities.

The process of setting the denominator to zero is used to find the values which cause the expression to be undefined. In the given expression \(\frac{1}{x-5} + 1\), we set the denominator \(x-5\) to zero and solve for x. The resulting key number, x = 5, indicates precisely at which point the denominator becomes zero and therefore, the expression cannot be evaluated. Students must remember to avoid these critical values in their calculations or when graphing rational expressions.