Chapter 8.5, Problem 50PPSE

Solution

To find: The simplest form of complex fraction $\frac{\frac{2}{x}-5}{\frac{6}{x}-3}$ .

The simplified form of complex fraction is written as $\frac{\frac{2}{x}-5}{\frac{6}{x}-3}=\frac{2-5x}{6-3x}$ where $x\ne 2$ . So, option $\left(A\right)$ is correct.

Given:

The complex fraction to simplify is $\frac{\frac{2}{x}-5}{\frac{6}{x}-3}$ .

Concept used:

Multiply the numerator and the denominator of the complex fraction by the least common multiple of numerator and denominator if required and then simplify the complex fraction.

Calculation:

By using the concept stated above, the complex fraction $\frac{\frac{2}{x}-5}{\frac{6}{x}-3}$ is simplified as:

  $\begin{array}{c}\frac{\frac{2}{x}-5}{\frac{6}{x}-3}=\frac{\frac{2}{x}-\frac{5}{1}}{\frac{6}{x}-\frac{3}{1}}\mathrm{....}\left(\text{Re-write}\right)\\ =\frac{\left(x\right)\left(\frac{2}{x}-\frac{5}{1}\right)}{\left(x\right)\left(\frac{6}{x}-\frac{3}{1}\right)}\mathrm{....}\left(\begin{array}{l}\text{Multiply numerator and }\\ \text{ denominator by }\left(x\right)\end{array}\right)\\ =\frac{2-5x}{6-3x}\mathrm{....}\left(\text{where }x\ne 2\right)\end{array}$

Conclusion:

The simplified form of complex fraction is written as $\frac{\frac{2}{x}-5}{\frac{6}{x}-3}=\frac{2-5x}{6-3x}$ where $x\ne 2$ . And, the simplified form of complex fraction matches with option $\left(A\right)\frac{2-5x}{6-3x}$ . So, option $\left(A\right)$ is correct.