Simplify; $\frac{a}{b} - \frac{b}{a} - \frac{c}{b}$

Analyze the question: We need to combine three fractions into a single fraction. The denominators are b, a and b, so the common denominator (LCD) is ab.

Concept: To add or subtract fractions, convert each to the same denominator by multiplying the top and bottom by what's missing.

Solution:

1. Convert each term to denominator ab: • a/b = (a · a)/(b · a) = a²/ab • b/a = (b · b)/(a · b) = b²/ab • c/b = (c · a)/(b · a) = ac/ab

2. Combine, keeping the signs:

a/b − b/a − c/b = (a² − b² − ac)/ab

3. Compare with the options: • A: a − b + c over ab, which is wrong because the terms were not converted properly. • B: ab − bc − ac over ab, which is wrong. • C: a² − b² + ac over ab, which has the wrong sign on ac. • D: a² − b² − ac over ab, which matches our result.

Tip: When subtracting fractions, put the whole numerator over the LCD and watch the signs. A minus sign in front of a fraction applies to its entire numerator. Check each option by testing simple numbers, for example a=2, b=1, c=1: the original gives 2 − 0.5 − 1 = 0.5, and D gives (4 − 1 − 2)/2 = 0.5 ✓.