Problem 5 - Introductory Algebra for College Students

Question

Find the least common denominator of the rational expressions. $$\frac{4}{x-3} \text { and } \frac{7}{x+1}$$

Step-by-Step Solution

Answer

The least common denominator of the fractional expressions \(\frac{4}{x-3}\) and \(\frac{7}{x+1}\) is \(x^2 -2x - 3\).

Step 1: Identify the Denominators

In the given rational expressions which are fractions, the denominators in these fractions are x-3 and x+1 respectively.

Step 2: Determine the Least Common Denominator

To find the least common denominator (LCD) of these expressions, since both denominators, x-3 and x+1 have no common factors other than 1, the LCD is simply the product of the two, which is (x-3)*(x+1). This is obtained by multiplying the two denominators together since there are no common factors.

Step 3: Simplify the Expression

To simplify this expression further, use the distributive property (or FOIL) where the first terms, outside terms, inside terms and last terms are multiplied. In this case, it would yield \(x^2 -3x + x -3 \), which simplified further becomes \(x^2 -2x - 3\)