If $\frac{2}{x - 3} - \frac{3}{x - 2}$ is equal to $\frac{p}{(x - 3)(x - 2)}$, find p

Analyze the question: We need to combine two fractions into a single fraction over the common denominator (x − 3)(x − 2) and read off the numerator p.

Concept: To subtract fractions, use the common denominator. Multiply each numerator by the factor missing from its denominator.

Solution:

2/(x − 3) − 3/(x − 2)

= [2(x − 2) − 3(x − 3)] / [(x − 3)(x − 2)]

Expand the numerator:

• 2(x − 2) = 2x − 4
• 3(x − 3) = 3x − 9

Numerator = (2x − 4) − (3x − 9) = 2x − 4 − 3x + 9 = −x + 5 = 5 − x

So p = 5 − x.

Check with a value: Let x = 4.

• Left side: 2/1 − 3/2 = 2 − 1.5 = 0.5
• Right side: p/(1·2) with p = 5 − 4 = 1 gives 1/2 = 0.5 ✓

Answer: D (5 − x)

Tip: The most common mistake is forgetting to distribute the minus sign over the second bracket: −3(x − 3) = −3x + 9, not −3x − 9. Always put brackets around the second numerator before expanding.