Questions: simplify the fraction [ fracfrac13+x-frac13x ]

Solution Steps

To simplify the given fraction, we need to perform the following steps:

1. Find a common denominator for the fractions in the numerator.
2. Subtract the fractions in the numerator.
3. Simplify the resulting expression by dividing by \(x\).

To simplify the expression \(\frac{\frac{1}{3+x} - \frac{1}{3}}{x}\), we first find a common denominator for the fractions in the numerator. The common denominator for \(\frac{1}{3+x}\) and \(\frac{1}{3}\) is \((3+x) \cdot 3\).

Using the common denominator, we rewrite the fractions: \[ \frac{1}{3+x} = \frac{3}{3(3+x)} \] \[ \frac{1}{3} = \frac{3+x}{3(3+x)} \] Subtracting these, we get: \[ \frac{3 - (3+x)}{3(3+x)} = \frac{-x}{3(3+x)} \]

Now, divide the result by \(x\): \[ \frac{\frac{-x}{3(3+x)}}{x} = \frac{-x}{3x(3+x)} \] Simplifying further, we cancel \(x\) in the numerator and denominator: \[ \frac{-1}{3(3+x)} \]

Final Answer