Problem 98 - Algebra and Trigonometry

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, since the evaluation of the expression \( \frac{3 x-3}{4 x(x-1)} \) at \( x=1 \) does not result in \( 0 \), but in an undefined mathematical operation, because it involves division by zero.

Step 1: Substitute \( x \) into the expression

Replace \( x \) with \( 1 \) in the expression: \( \frac{3 x-3}{4 x(x-1)} \) becomes \( \frac{3(1)-3}{4(1)(1-1)} \)

Step 2: Simplify the substituted expression

The simplified expression becomes: \( \frac{3-3}{0} \). However, as per the laws of mathematics, division by zero is undefined.

Step 3: Verify the claim

In the original statement, it says the outcome of the evaluation is \( 0 \). However, the denominator turns to zero when \( x = 1 \) and this leads to an undefined condition, not a zero.

Key Concepts

Undefined Expressions

In mathematics, an expression becomes undefined when we encounter situations like division by zero. Division by zero is a rule that's strictly enforced because dividing a number by zero does not produce a valid result. Let's break it down:

• Any number divided by zero does not have a defined value. The result is not zero; rather, it's undefined.
• When an expression contains a division with zero as the denominator, we can't calculate a valid numeric result.
• In the case of our exercise, substituting \(x = 1\) into \(\frac{3x-3}{4x(x-1)}\) resulted in the denominator becoming zero. This reveals why the expression is undefined rather than zero.

Recognizing these scenarios is fundamental for solving algebraic problems correctly. Always be on the lookout for conditions where the denominator might turn zero during calculations.

Substitution in Algebra

Substitution in algebra is a basic method for solving and simplifying expressions or equations. It involves replacing a variable with a specific value to evaluate the expression. Here’s how you effectively use substitution:

• Identify the variable and the value to substitute in the expression or equation.
• Replace each instance of the variable with the given value throughout the expression.
• Proceed to simplify the new expression by following the order of operations, such as performing operations within parentheses first, then multiplication or division, and finally addition or subtraction.

In our exercise, substituting \(x = 1\) was performed on the expression \(\frac{3x-3}{4x(x-1)}\). By substituting 1 for \(x\), we discovered a zero in the denominator, leading to an undefined expression. Ensure to consider how substitutions affect the parts of your expression when working through your own algebra problems.

Fraction Simplification

Fraction simplification is about reducing fractions to their simplest form, where the greatest common factor (GCF) of the numerator and the denominator is 1. However, care must be taken when simplifying fractions containing variables:

• Start by factoring both the numerator and the denominator. Look for common factors, which can be constants or expressions containing variables.
• Cancel out the common factors in both parts of the fraction, but take note not to cancel terms incorrectly.
• After cancelation, ensure no further simplification is possible.

It's important to recognize that simplification can always be performed unless restricted by zero in the denominator. In the task, simplifying \(\frac{3-3}{0}\) was attempted, but the fraction becomes undefined due to zero in the denominator. Always verify that simplification steps don’t inadvertently cause undefined expressions, especially when dealing with variable-dependent denominators.