Question

the image shows a complex fraction: the numerator is (x + 4)/3 + 1/x, and the denominator is 3/x² - 1/3.

Explanation:

To simplify the complex fraction \(\frac{\frac{x + 4}{3}+\frac{1}{x}}{\frac{3}{x^{2}}-\frac{1}{3}}\), we follow these steps:

Step 1: Simplify the numerator

The numerator is \(\frac{x + 4}{3}+\frac{1}{x}\). To add these two fractions, we find a common denominator, which is \(3x\).

\[

$$\begin{align*} \frac{x + 4}{3}+\frac{1}{x}&=\frac{(x + 4)x}{3x}+\frac{3}{3x}\\ &=\frac{x(x + 4)+3}{3x}\\ &=\frac{x^{2}+4x + 3}{3x} \end{align*}$$

\]

We can factor the numerator \(x^{2}+4x + 3\) as \((x + 1)(x + 3)\), so the numerator simplifies to \(\frac{(x + 1)(x + 3)}{3x}\).

Step 2: Simplify the denominator

The denominator is \(\frac{3}{x^{2}}-\frac{1}{3}\). To subtract these two fractions, we find a common denominator, which is \(3x^{2}\).

\[

$$\begin{align*} \frac{3}{x^{2}}-\frac{1}{3}&=\frac{9}{3x^{2}}-\frac{x^{2}}{3x^{2}}\\ &=\frac{9 - x^{2}}{3x^{2}} \end{align*}$$

\]

We can factor the numerator \(9 - x^{2}\) as a difference of squares: \(9 - x^{2}=(3 - x)(3 + x)\), so the denominator simplifies to \(\frac{(3 - x)(3 + x)}{3x^{2}}\).

Step 3: Divide the simplified numerator by the simplified denominator

Dividing the simplified numerator \(\frac{(x + 1)(x + 3)}{3x}\) by the simplified denominator \(\frac{(3 - x)(3 + x)}{3x^{2}}\) is equivalent to multiplying the numerator by the reciprocal of the denominator:

\[

$$\begin{align*} \frac{\frac{(x + 1)(x + 3)}{3x}}{\frac{(3 - x)(3 + x)}{3x^{2}}}&=\frac{(x + 1)(x + 3)}{3x}\times\frac{3x^{2}}{(3 - x)(3 + x)}\\ &=\frac{(x + 1)(x + 3)\times3x^{2}}{3x\times(3 - x)(3 + x)} \end{align*}$$

\]

Step 4: Cancel common factors

We can cancel the common factors of \(3\), \(x\), and \((x + 3)\) (note that \((x + 3)\) and \((3 + x)\) are the same):

\[

$$\begin{align*} \frac{(x + 1)(x + 3)\times3x^{2}}{3x\times(3 - x)(3 + x)}&=\frac{(x + 1)\times\cancel{(x + 3)}\times\cancel{3x^{2}}}{\cancel{3x}\times(3 - x)\times\cancel{(3 + x)}}\\ &=\frac{x(x + 1)}{3 - x} \end{align*}$$

\]

Final Answer

\(\boxed{\frac{x(x + 1)}{3 - x}}\)

Answer:

To simplify the complex fraction \(\frac{\frac{x + 4}{3}+\frac{1}{x}}{\frac{3}{x^{2}}-\frac{1}{3}}\), we follow these steps:

Step 1: Simplify the numerator

The numerator is \(\frac{x + 4}{3}+\frac{1}{x}\). To add these two fractions, we find a common denominator, which is \(3x\).

\[

$$\begin{align*} \frac{x + 4}{3}+\frac{1}{x}&=\frac{(x + 4)x}{3x}+\frac{3}{3x}\\ &=\frac{x(x + 4)+3}{3x}\\ &=\frac{x^{2}+4x + 3}{3x} \end{align*}$$

\]

We can factor the numerator \(x^{2}+4x + 3\) as \((x + 1)(x + 3)\), so the numerator simplifies to \(\frac{(x + 1)(x + 3)}{3x}\).

Step 2: Simplify the denominator

The denominator is \(\frac{3}{x^{2}}-\frac{1}{3}\). To subtract these two fractions, we find a common denominator, which is \(3x^{2}\).

\[

$$\begin{align*} \frac{3}{x^{2}}-\frac{1}{3}&=\frac{9}{3x^{2}}-\frac{x^{2}}{3x^{2}}\\ &=\frac{9 - x^{2}}{3x^{2}} \end{align*}$$

\]

We can factor the numerator \(9 - x^{2}\) as a difference of squares: \(9 - x^{2}=(3 - x)(3 + x)\), so the denominator simplifies to \(\frac{(3 - x)(3 + x)}{3x^{2}}\).

Step 3: Divide the simplified numerator by the simplified denominator

Dividing the simplified numerator \(\frac{(x + 1)(x + 3)}{3x}\) by the simplified denominator \(\frac{(3 - x)(3 + x)}{3x^{2}}\) is equivalent to multiplying the numerator by the reciprocal of the denominator:

\[

$$\begin{align*} \frac{\frac{(x + 1)(x + 3)}{3x}}{\frac{(3 - x)(3 + x)}{3x^{2}}}&=\frac{(x + 1)(x + 3)}{3x}\times\frac{3x^{2}}{(3 - x)(3 + x)}\\ &=\frac{(x + 1)(x + 3)\times3x^{2}}{3x\times(3 - x)(3 + x)} \end{align*}$$

\]

Step 4: Cancel common factors

We can cancel the common factors of \(3\), \(x\), and \((x + 3)\) (note that \((x + 3)\) and \((3 + x)\) are the same):

\[

$$\begin{align*} \frac{(x + 1)(x + 3)\times3x^{2}}{3x\times(3 - x)(3 + x)}&=\frac{(x + 1)\times\cancel{(x + 3)}\times\cancel{3x^{2}}}{\cancel{3x}\times(3 - x)\times\cancel{(3 + x)}}\\ &=\frac{x(x + 1)}{3 - x} \end{align*}$$

\]

Final Answer

\(\boxed{\frac{x(x + 1)}{3 - x}}\)