Problem 98 - College Algebra

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I evaluated \(\frac{3 x-3}{4 x(x-1)}\) for \(x-1\) and obtained 0.

Step-by-Step Solution

Answer

The statement does not make sense because replacing \(x\) with \(x-1\) modifies the function itself and one can't expect the same outcome as the original function. In fact, after replacing \(x\) with \(x-1\) and simplifying, the function changes to \(\frac{3x-6}{4x^2-8x+4}\), which can't yield 0 for the same values that would do so in the original function.

Step 1: Understand the Statement

The statement is saying that \(\frac{3 x-3}{4 x(x-1)}\) was evaluated with \(x-1\) replacing \(x\) and 0 was obtained. If this substitution is done correctly, you would get \(\frac{3(x-1)-3}{4*(x-1)*(x-1-1)}\), which creates an entirely different expression.

Step 2: Evaluate the new expression

Evaluate the new expression \(\frac{3(x-1)-3}{4*(x-1)*(x-1-1)}\) by distributing and simplifying. This would simplify to \(\frac{3x-6}{4x^2-8x+4}\), which is different from the original function.

Step 3: Determine if the statement makes sense

The statement does not make sense because substituting \(x-1\) in place of \(x\) changes the function itself, which means that it should not yield the same result as the original function.

Key Concepts

Substitution Method

The substitution method is a fundamental technique used in mathematics, especially in algebra, where a specific value or expression replaces a variable in an equation. This method is versatile and helps to simplify equations, evaluate expressions, or even solve systems of equations. In the context of rational expressions, like the one in our exercise, substitution involves plugging in a particular value for the variable. For instance, replacing the variable 'x' with another expression or value transforms the equation accordingly. Let's see a few key points:

• Purpose of Substitution: It simplifies calculations by allowing evaluation with specific values, and to check the validity of certain values within an equation.


• Process: Carefully replace each occurrence of the variable with the new expression or value. This can lead to a simpler equation or help in finding unknown values.


• Checking Consistency: After substitution, it's important to verify if the new expression holds the same properties as the original, e.g., does it still provide the same result?

The substitution method helps in analyzing whether transforming a rational expression maintains its integrity or not, as shown in the provided exercise.

Simplifying Expressions

Simplifying expressions is a key algebraic skill that involves reducing expressions to their simplest form. This is done through various mathematical operations that often involve removing parentheses, combining like terms, or canceling common factors. When working with rational expressions, simplifying helps in reducing complexity and identifying any errors in evaluation.

• Simplification Process: It often begins by expanding expressions, where you distribute multiplication over addition or subtraction.


• Cancel Common Factors: If there is a common factor in both the numerator and the denominator, it can be canceled out to simplify the expression.


• Example from Exercise: When substituting and simplifying \(\frac{3(x-1)-3}{4*(x-1)*(x-1-1)}\), we expand the terms in the numerator and denominator to eventually simplify to \(\frac{3x-6}{4x^2-8x+4}\).

Simplifying expressions helps in understanding the new structure created after substitution and is essential for further algebraic evaluation.

Algebraic Evaluation

Algebraic evaluation is the process of solving or finding the value of an expression once variables have been substituted with actual numbers. This process tests the understanding of mathematical operations and the ability to correctly follow order of operations—often referred to as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).In rational expressions, consider these points:

• Evaluate Step-by-Step: Plug in values for variables, simplify expressions, and calculate results systematically.


• Verify Outcomes: After finding the result, check if it logically makes sense. Verify by simplifying the expression again to avoid calculation errors.


• Exercise Context: When evaluating the modified expression \(\frac{3x-6}{4x^2-8x+4}\), it reveals whether the statement about it yielding zero is reasonable or not.

Understanding how to properly evaluate algebraic expressions by substitution and simplification ensures clarity and accuracy in mathematical conclusions, just as demonstrated in the exercise.