Question

How do you write as a single fraction in its simplest form: \[\dfrac{5}{{2 - x}} - \dfrac{4}{x}\]?

Answer 1

Hint: Here is the given question in the form of subtraction of two fractions. Observe the denominator of both the fractions there are unlike fractions before going to subtract this first make two fractions are like or using a method of Cross multiplication i.e., \[\dfrac{A}{B} - \dfrac{C}{D} = \dfrac{{AD - BC}}{{BD}}\]. On further simplification we get the required solution in a single fraction.

Complete step-by-step answer:
In the given two fractions are unlike fractions.
When we add and subtract two unlike fractions, we have to make the denominator equal first and then perform the respective operation. There are two methods by which we can make the denominator equal. They are: Cross-Multiplication Method and LCM Method
In the cross multiplication method, we cross multiply the numerator of the first fraction by the denominator of the second fraction. Then multiply the numerator of the second fraction by the denominator of the first fraction. Now, multiply both the denominators and take it as a common denominator. Then we can add or subtract the fractions now. i.e.,
\[\dfrac{A}{B} + \dfrac{C}{D} = \dfrac{{AD + BC}}{{BD}}\] and \[\dfrac{A}{B} - \dfrac{C}{D} = \dfrac{{AD - BC}}{{BD}}\]
Let consider the given fraction
\[ \Rightarrow \dfrac{5}{{2 - x}} - \dfrac{4}{x}\]
Using the cross multiplication method
\[ \Rightarrow \dfrac{{5x - 4\left( {2 - x} \right)}}{{x\left( {2 - x} \right)}}\]
\[ \Rightarrow \dfrac{{5x - 8 + 4x}}{{x\left( {2 - x} \right)}}\]
\[ \Rightarrow \dfrac{{9x - 8}}{{x\left( {2 - x} \right)}}\]
Hence the simplest form of single fraction of the fraction \[\dfrac{5}{{2 - x}} - \dfrac{4}{x}\] is \[\dfrac{{9x - 8}}{{x\left( {2 - x} \right)}}\].
So, the correct answer is “\[\dfrac{{9x - 8}}{{x\left( {2 - x} \right)}}\]”.

Note: In the fraction usually we take LCM to solve. Where LCM is abbreviated as least common factor. But when the denominator value contains a variable, we have used cross multiplication, then we simplify the given fraction. while multiplying we should take care of the signs of the numerals and variables.