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Chapter 0: Problem 44

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

Short Answer

Expert verified
The result of the subtraction \(\frac{4}{x}-\frac{3}{x+3}\) is \(\frac{x + 12}{x^2 + 3x}\).

Step by step solution

01

Identify the LCD

First, let's find the least common denominator (LCD), which is the smallest number that both denominators divide evenly into. Since the denominators are not constants, but expressions involving \(x\), we must consider potential values of \(x\) that won't make the denominators equal to zero. For the fractions \(\frac{4}{x}\) and \(\frac{3}{x+3}\), the LCD will be \(x \cdot (x+3)\).
02

Rewrite the fractions with the LCD

Next, rewrite each fraction such that each has the LCD, \(x \cdot (x+3)\), as its denominator. Because we multiplied the denominator of \(\frac{4}{x}\) by \((x+3)\), we do the same for its numerator, giving us \(\frac{4 \cdot (x+3)}{x \cdot (x+3)}\). Similarly, because we multiplied the denominator of \(\frac{3}{x + 3}\) by \(x\), we do the same for its numerator, resulting in \(\frac{3x}{x \cdot (x+3)}\).
03

Subtract the fractions

Now with the same denominators, we can subtract the fractions. So, \(\frac{4 \cdot (x+3)}{x \cdot (x+3)} - \frac{3x}{x \cdot (x+3)} = \frac{4 \cdot (x+3) - 3x}{x \cdot (x+3)}\).
04

Simplify the result

Simplify the result by multiplying out and combine like terms in the numerator to get \(\frac{4x + 12 - 3x}{x^2 + 3x}\), which simplifies further to \(\frac{x + 12}{x^2 + 3x}\).

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