Question

\(\frac{1-x}{x-1} = \frac{1}{x}, \; x \neq 1\)

Column A	Column B
\(x\)	\(-\frac{1}{2}\)

Hint:

A common algebraic trick is recognizing expressions of the form \(\frac{a-b}{b-a}\). As long as \(a \neq b\), this fraction always simplifies to -1. Spotting this pattern instantly solves the equation.

Answer 1

Step 1: Understanding the Concept:
This question requires solving an algebraic equation for the variable \(x\) and then comparing the result to a given fraction.
Step 2: Key Formula or Approach:
The key to solving the equation is to recognize the relationship between the numerator and denominator of the fraction on the left side. The term \((1-x)\) is the negative of \((x-1)\).
Step 3: Detailed Explanation:
The equation is \(\frac{1-x}{x-1} = \frac{1}{x}\).
Let's simplify the left side. We can factor -1 out of the numerator:
\[ 1-x = -(-1+x) = -(x-1) \] Substitute this back into the equation:
\[ \frac{-(x-1)}{x-1} = \frac{1}{x} \] Since we are given that \(x \neq 1\), the term \((x-1)\) is not zero, so we can cancel it from the numerator and denominator:
\[ -1 = \frac{1}{x} \] To solve for \(x\), multiply both sides by \(x\):
\[ -x = 1 \] Multiply by -1 to get the final value for \(x\):
\[ x = -1 \] Now we compare the columns.
Column A: \(x = -1\).
Column B: \(-\frac{1}{2}\).
On a number line, -1 is to the left of \(-\frac{1}{2}\). Therefore, -1 is less than \(-\frac{1}{2}\).
Step 4: Final Answer:
The value of \(x\) is -1. Since \(-1<-\frac{1}{2}\), the quantity in Column B is greater.