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Question

Difficulty: MediumRational and Radical Expressions and Equations

For all real numbers xx such that x≠−2x \neq -2 and x≠3x \neq 3, the expression 2xx−3−5x+2\frac{2x}{x - 3} - \frac{5}{x + 2} is equivalent to which of the following?

  1. A
    2x−52x−1\frac{2x - 5}{2x - 1}
  2. B
    \frac{2x^2 - x - 15}{x^2 - x - 6}
  3. C
    \frac{x + 15}{x^2 - x - 6}
  4. \frac{2x^2 - x + 15}{x^2 - x - 6}Answer
  5. E
    \frac{2x^2 - x + 15}{x^2 - 6}

Answer

The expression is equivalent to 2x2−x+15x2−x−6\frac{2x^2 - x + 15}{x^2 - x - 6}.
To subtract the rational expressions, we find the common denominator, which is (x−3)(x+2)=x2−x−6(x - 3)(x + 2) = x^2 - x - 6. We rewrite each fraction with this denominator, which gives 2x(x+2)x2−x−6=2x2+4xx2−x−6\frac{2x(x + 2)}{x^2 - x - 6} = \frac{2x^2 + 4x}{x^2 - x - 6} and 5(x−3)x2−x−6=5x−15x2−x−6\frac{5(x - 3)}{x^2 - x - 6} = \frac{5x - 15}{x^2 - x - 6}. Subtracting the second numerator from the first gives (2x2+4x)−(5x−15)=2x2−x+15(2x^2 + 4x) - (5x - 15) = 2x^2 - x + 15. Placing this over the common denominator gives the simplified expression 2x2−x+15x2−x−6\frac{2x^2 - x + 15}{x^2 - x - 6}.

Step-by-Step Solution

1
Identify the common denominator.
The common denominator is (x−3)(x+2)=x2−x−6(x - 3)(x + 2) = x^2 - x - 6.
To subtract rational expressions, we need a common denominator.
2
Rewrite each rational expression with the common denominator.
The first term becomes 2x(x+2)(x−3)(x+2)=2x2+4xx2−x−6\frac{2x(x + 2)}{(x - 3)(x + 2)} = \frac{2x^2 + 4x}{x^2 - x - 6}, and the second term becomes 5(x−3)(x−3)(x+2)=5x−15x2−x−6\frac{5(x - 3)}{(x - 3)(x + 2)} = \frac{5x - 15}{x^2 - x - 6}.
Multiplying the numerator and denominator of each term by the missing factor keeps the values of the expressions unchanged.
3
Subtract the numerators.
(2x2+4x)−(5x−15)=2x2+4x−5x+15=2x2−x+15(2x^2 + 4x) - (5x - 15) = 2x^2 + 4x - 5x + 15 = 2x^2 - x + 15.
Subtracting the second numerator requires distributing the negative sign to both terms of the expression (5x−15)(5x - 15).
4
Combine the result over the common denominator.
\frac{2x^2 - x + 15}{x^2 - x - 6}
Write the simplified numerator over the common denominator.

Key Concept

Subtraction of rational expressions involves finding a common denominator, expanding the numerators, and distributing negative signs carefully.
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Topics

Rational and Radical Expressions and Equations

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