Which expression is equivalent to 2/(3 - 2x) + 1/(x + 3)?\(\frac{1}{(3 - 2\mathrm{x})(\mathrm{x} + 3)}\)3/(4x - 6)\(\frac{9}{(2\mathrm{x} - 3)(\mathrm...

1. INFER the strategy needed

• We need to add two fractions: \(\frac{2}{3 - 2\mathrm{x}} + \frac{1}{\mathrm{x} + 3}\)
• To add fractions, we need a common denominator
• The common denominator will be \((3 - 2\mathrm{x})(\mathrm{x} + 3)\)

2. SIMPLIFY each fraction to have the common denominator

• First fraction: \(\frac{2}{3 - 2\mathrm{x}} = \frac{2(\mathrm{x} + 3)}{(3 - 2\mathrm{x})(\mathrm{x} + 3)} = \frac{2\mathrm{x} + 6}{(3 - 2\mathrm{x})(\mathrm{x} + 3)}\)
• Second fraction: \(\frac{1}{\mathrm{x} + 3} = \frac{1(3 - 2\mathrm{x})}{(3 - 2\mathrm{x})(\mathrm{x} + 3)} = \frac{3 - 2\mathrm{x}}{(3 - 2\mathrm{x})(\mathrm{x} + 3)}\)

3. SIMPLIFY by adding the fractions

• Since both fractions have the same denominator, add the numerators:
• \(\frac{2\mathrm{x} + 6}{(3 - 2\mathrm{x})(\mathrm{x} + 3)} + \frac{3 - 2\mathrm{x}}{(3 - 2\mathrm{x})(\mathrm{x} + 3)} = \frac{2\mathrm{x} + 6 + 3 - 2\mathrm{x}}{(3 - 2\mathrm{x})(\mathrm{x} + 3)}\)

4. SIMPLIFY the numerator

• \(2\mathrm{x} + 6 + 3 - 2\mathrm{x} = 9\) (the 2x terms cancel out)

Answer: \(\frac{9}{(3 - 2\mathrm{x})(\mathrm{x} + 3)}\) = Choice D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make algebraic mistakes when converting fractions to the common denominator or when combining the numerators. They might incorrectly distribute signs or make errors combining like terms, leading to a numerator that isn't 9. This could cause them to select Choice A (\(\frac{1}{(3 - 2\mathrm{x})(\mathrm{x} + 3)}\)) or Choice B (\(\frac{3}{4\mathrm{x} - 6}\)).

Second Most Common Error:

Sign confusion: Students might confuse \((3 - 2\mathrm{x})\) with \((2\mathrm{x} - 3)\), especially when looking at the answer choices. Since these expressions are negatives of each other, this confusion could lead them to select Choice C (\(\frac{9}{(2\mathrm{x} - 3)(\mathrm{x} + 3)}\)) instead of the correct answer.

The Bottom Line:

This problem tests your ability to systematically add rational expressions. The key is staying organized with the algebra and being careful with signs when working with expressions like \((3 - 2\mathrm{x})\).