Simplify; $\frac{2}{1 - x} - \frac{1}{x}$

Analyze the question: We need to combine two fractions with different denominators, (1 − x) and x, into a single fraction.

Concept: To subtract fractions, find a common denominator. Here it is x(1 − x). Multiply each fraction's numerator by whatever is missing from its denominator.

Solution:

Expand the numerator:

So:

Check with a value: Let x = 2.

• Original: 2/(1−2) − 1/2 = −2 − 0.5 = −2.5
• Option B: (3(2) − 1)/(2(1−2)) = 5/(−2) = −2.5 ✓

Answer: Option B.

Tip: The most common mistake is forgetting that the minus sign applies to the whole of (1 − x), which gives −1 + x, not −1 − x. Always put brackets around the numerator being subtracted. A quick check by substituting a number such as x = 2 can confirm your answer in seconds.