Problem 62 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{2 x-1}{x+6}-\frac{6-5 x}{x^{2}-36}$$

Step-by-Step Solution

Answer

The result of the operation \(\frac{2 x-1}{x+6}-\frac{6-5 x}{x^{2}-36}\) is \(\frac{-3x^2+19x-12}{(x+6)(x-6)}\).

Step 1: Simplify the Fractions

We have two fractions \(\frac{2x-1}{x+6}\) and \(\frac{6-5x}{x^{2}-36}\). Our aim is to simplify the second fraction. As \(x^{2}-36\) can be factored as \((x-6)(x+6)\), this gives us two fractions \(\frac{2x-1}{x+6}\) and \(\frac{6-5x}{(x-6)(x+6)}\).

Step 2: Find common denominator

The next step is to find a common denominator so that we can add or subtract the two fractions. The common denominator of \(\frac{2x-1}{x+6}\) and \(\frac{6-5x}{(x-6)(x+6)}\) is \((x+6)(x-6)\). So, we should multiply the numerator and denominator of the first fraction by \(x-6\). This gives us \(\frac{(2x-1)(x-6)}{(x+6)(x-6)}\) - \(\frac{6-5x}{(x-6)(x+6)}.\)

Step 3: Subtract the fractions

Next, subtract \(\frac{6-5x}{(x-6)(x+6)}\) from \(\frac{(2x-1)(x-6)}{(x+6)(x-6)}\). This results in \(\frac{(2x-1)(x-6)-(6-5x)}{(x+6)(x-6)}\).

Step 4: Simplify the result

Now, simplify the equation and expand the numerator \((2x-1)(x-6)-(6-5x)\) which gives \(-3x^2+19x-12\). Thus we get \(\frac{-3x^2+19x-12}{(x+6)(x-6)}\). The result can't be simplified further.