Problem 49 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{2 x+9}{x^{2}-7 x+12}-\frac{2}{x-3}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{17}{(x-3)(x-4)}\)

Step 1: Simplify the denominator in the first fraction

The denominator in the first fraction, \(x^{2}-7x+12\), can be factored. After factoring, it becomes \((x-3)(x-4)\)

Step 2: Find a common denominator

In the second fraction, the denominator is \(x-3\), and in the first, it is \((x-3)(x-4)\). The common denominator here is \((x-3)(x-4)\). The second fraction can be rewritten over this denominator as \(\frac{2(x-4)}{(x-3)(x-4)}\)

Step 3: Subtract the fractions

Once the fractions have a common denominator, they can be subtracted. \(\frac{(2x+9)-(2x-8)}{(x-3)(x-4)}\). Simplifying this expression gives \(\frac{17}{(x-3)(x-4)}\)