Problem 50 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{3 x+7}{x^{2}-5 x+6}-\frac{3}{x-3}$$

Step-by-Step Solution

Answer

\(\frac{13}{(x-3)(x-2)}\)

Step 1: Factor the Denominator

Factor the denominator of the first fraction \(x^{2}-5 x+6\) to \((x-3)(x-2)\). So, the fraction can be written as \(\frac{3 x+7}{(x-3)(x-2)}\).

Step 2: Identify Common Denominator

The denominators of the two expressions are \((x-3)(x-2)\) and \(x-3\). The least common denominator (LCD) is \( (x-3)(x-2)\). So rewrite the subtracted fraction with this denominator as \(\frac{3(x-2)}{(x-3)(x-2)}\). The expression then becomes \(\frac{3 x+7}{(x-3)(x-2)}-\frac{3(x-2)}{(x-3)(x-2)}\).

Step 3: Subtract Numerators

Since the fractions have the same denominator, subtract the numerators: \(\frac{3 x+7-3(x-2)}{(x-3)(x-2)} = \frac{3x+7-3x+6}{(x-3)(x-2)} = \frac{13}{(x-3)(x-2)}\) (You can verify that the numerator can't be simplified further by performing the polynomial subtraction).

Step 4: Final Answer

Thus, the expression is simplified to \(\frac{13}{(x-3)(x-2)}\).