Chapter 6.6, Problem 30WE

To determine

Simplify: the given expression as fraction.

Answer to Problem 30WE

The simplified form of $\left(x+4\right)\left(\frac{4}{x}-1\right)$ is $\frac{\left(4-x\right)\left(4+x\right)}{x}$.

Explanation of Solution

Given information:

The expression $\left(x+4\right)\left(\frac{4}{x}-1\right)$.

Formula used:

Least common denominator: It is the smallest multiple of denominator of set of fractions.

  $a^2-b^2=\left(a+b\right)\left(a-b\right)$

  $\begin{array}{l}\left(x+4\right)\left(\frac{4}{x}-1\right)\\ =x\cdot \frac{4}{x}-x\cdot 1+4\cdot \frac{4}{x}-4\cdot 1\\ =4-x+\frac{16}{x}-4\\ =\frac{16}{x}-x\end{array}$

Recall that Least common denominator is the smallest multiple of denominator of set of fractions.

  $=\frac{16}{x}-\frac{x}{1}$

Least common denominator for $\frac{16}{x},\frac{x}{1}$ is $x$ . So,

  $\begin{array}{l}=\frac{16-x^2}{x}\\ =\frac{4^2-x^2}{x}\end{array}$

Factorizing the numerator of expression by using identity $a^2-b^2=\left(a+b\right)\left(a-b\right)$.

  $=\frac{\left(4-x\right)\left(4+x\right)}{x}$

The simplified form of $\left(x+4\right)\left(\frac{4}{x}-1\right)$ is $\frac{\left(4-x\right)\left(4+x\right)}{x}$.