Problem 43 - College Algebra

Question

Add or subtract as indicated. $$\frac{3}{x+1}-\frac{3}{x}$$

Step-by-Step Solution

Answer

The answer is \(-\frac{3}{x^2+x}\)

Step 1: Find the Common Denominator

The common denominator in this case is the product of the two denominators \(x\) and \(x+1\). Which results to \(x(x+1)\).

Step 2: Convert the Fractions to have the Same Denominator

The given fractions can be rewritten with the common denominator. The first fraction can be rewritten as \(\frac{3x}{x(x+1)}\), and the second fraction can be rewritten as \(\frac{3(x+1)}{x(x+1)}\). This is achieved by multiplying the numerator and denominator of each fraction by the factor that will give the common denominator.

Step 3: Execute the Subtraction

Now that the fractions have the same denominator, they can be subtracted. Hence, the expression becomes \(\frac{3x - 3(x+1)}{x(x+1)}\). This means that we subtract the numerators and keep the denominator the same.

Step 4: Simplify the Numerator

Simplify the numerator to give \(\frac{3x - 3x -3}{x(x+1)}\), which reduces to \(\frac{-3}{x(x+1)}\). Conversely, we can also simplify the original subtraction as \(\frac{3x - 3x -3}{x^2+x}\).