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Chapter 0: Problem 43

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

Short Answer

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

Step by step solution

01

Find the Common Denominator

The common denominator in this case is the product of the two denominators \(x\) and \(x+1\). Which results to \(x(x+1)\).
02

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.
03

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.
04

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 \(\frac{3x - 3x -3}{x^2+x}\).

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