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Chapter 0: Problem 44

Add or subtract as indicated. $$\frac{4}{x}-\frac{3}{x+3}$$

Short Answer

Expert verified
So, \(\frac{4}{x} - \frac{3}{x + 3}\) simplifies to \(\frac{x + 12}{x^2 + 3x}\).

Step by step solution

01

Identify the common denominator

The denominators of the two fractions are \(x\) and \(x + 3\). To combine these fractions, a common denominator is needed. A simple method to find a common denominator is to multiply the two different denominators. So, the common denominator in this case would be \(x * (x + 3)\) or \(x^2 + 3x\).
02

Rewrite the fractions

Next, rewrite each fraction with the common denominator of \(x^2 + 3x\). For the first fraction \(\frac{4}{x}\), multiply both the numerator and denominator by \(x + 3\). This gives \(\frac{4(x + 3)}{x^2 + 3x}\). For the second fraction \(\frac{3}{x + 3}\), multiply both the numerator and denominator by \(x\). This gives \(\frac{3x}{x^2 + 3x}\).
03

Subtraction of the fractions

We can now substitute the two new fractions back, replacing the original ones in the expression. This results to \(\frac{4(x + 3)}{x^2 + 3x} - \frac{3x}{x^2 + 3x}\). Since they have the same denominator, they can be combined into one fraction: \(\frac{4(x + 3) - 3x}{x^2 + 3x}\).
04

Simplify the fraction

We can further simplify the fraction by expanding the numerator and combining like terms. This gives: \(\frac{4x + 12 - 3x}{x^2 + 3x} = \frac{x + 12}{x^2 + 3x}\).

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