Problem 26 - Prealgebra and Introductory Algebra

Question

True or false? \(\frac{3}{x-8}+\frac{3}{8-x}=0\)

Step-by-Step Solution

Answer

The statement is true for all real numbers except \(x=8\).

Step 1: Simplify the equation

Firstly, observe that \(x-8\) and \(8-x\) are negative inverses of each other. It means that \(x-8=-(8-x)\), and the signs of the denominators can be easily reversed. Reversing the sign of \(8-x\) will give us \(x-8\) which results in the equation becoming: \(\frac{3}{x-8}-\frac{3}{x-8}=0\)

Step 2: Combine the fractions

Combining the fractions with the same denominators \(\frac{3}{x-8}-\frac{3}{x-8}\) will result in \(0\)

Step 3: Final step

In the end the proposed equation is indeed \(0\), hence it's a true statement. However, it's important to note that this is only valid for all \(x\neq 8\) to avoid dividing by zero

Key Concepts

Fraction Simplification

Fraction simplification involves rewriting fractions to their simplest form, making computations easier.
This often means reducing the fraction's numerator and denominator to their smallest possible integers while still having the same ratio.To simplify, find the greatest common divisor (GCD) of both the numerator and denominator, and divide both by this number.

• If you're simplifying a fraction like \( \frac{6}{8} \), you'll find the GCD of 6 and 8 is 2.
• Dividing both the numerator and denominator by 2, you get \( \frac{3}{4} \).

Simplifying doesn't always involve whole numbers. In the exercise, observing negative inverses is a sophisticated method of simplification.Understanding fraction simplification is essential as it often connects to other concepts like combining fractions.

Negative Inverses

Understanding negative inverses can vastly simplify algebraic fractions. This concept deals with terms that are opposites of each other.
When you notice expressions like \( x - 8 \) and \( 8 - x \), realize they are negative inverses.
Algebraically, \( x - 8 \) can be transformed to \( -(8 - x) \), allowing simplification. When simplified in equations, reversing the sign of one term helps in standardizing fractions.

• Reversing signs can result in identical denominators, aiding in combining fractions.
• It simplifies the process, streamlining the solution.

Mastering negative inverses provides agility in tackling otherwise complex algebraic problems, ensuring you are prepared for equations that may appear challenging at first glance.

Combining Fractions

Combining fractions is common in algebra, especially when they share the same denominator.
With a shared denominator, you add or subtract the numerators while the denominator remains consistent.
For the example \( \frac{3}{x-8} - \frac{3}{x-8} \), you directly subtract the numerators, leading to zero.

• Only worry about numerators if denominators are identical.
• Combine through addition or subtraction depending on the signs.

Sometimes, making denominators the same involves simplification or manipulation using negative inverses. Effectively combining fractions not only simplifies equations but also prevents errors in longer, more complex problems. When facing equations or expressions with fractions, ensuring a unified denominator is crucial.