Problem 67 - College Algebra Essentials

Question

Simplify each complex rational expression. $$\frac{\frac{3}{x-2}-\frac{4}{x+2}}{\frac{7}{x^{2}-4}}$$

Step-by-Step Solution

Answer

\(\frac{-x+14}{7}\)

Step 1: Simplify the numerator

In order to deal effectively with the expression, it is necessary to simplify it, starting with the numerator. This will be done by getting rid of the denominators \(x-2\) and \(x+2\) in the numerator. To achieve this, multiply the first term by \((x+2)/(x+2)\) and the second term by \((x-2)/(x-2)\). Doing this, our expression becomes \((((3(x+2) - 4(x-2))/((x-2)(x+2)))/\frac{7}{x^{2}-4})\), which simplifies to \(\frac{(3x+6 - 4x+8) / (x^{2} - 4)}{\frac{7}{x^{2}-4}}\), further simplifying to \(\frac{-x+14}{x^{2} - 4}\) divided by \(\frac{7}{x^{2} - 4}\)

Step 2: Convert the division to multiplication

Dealing with division can be difficult, so it's helpful to convert the division operation to multiplication, which simplifies things. We can do this by multiplying the first fraction by the RECIPROCAL of the second fraction. That is \(\frac{-x+14}{x^{2}-4} * \frac{x^{2}-4}{7}\)

Step 3: Simplify the expression

Now, we can simplify the expression by canceling out the like terms which are present in the numerator and the denominator of the expression. In this case, \(x^{2} - 4\) is present in both, so we can cancel them out. Doing this simplification, we get \(\frac{-x+14}{7}\) as the final simplified complex rational expression.