Question

How do you simplify \[\dfrac{{x + 5}}{{x - 5}} \div \dfrac{{{x^2} - 25}}{{5 - x}}\]?

Answer 1

Hint: We have to simplify the given expression. As we know, dividing a fraction by another fraction is the same as multiplying the fraction by the reciprocal of the other. So, we will write the given expression as \[\dfrac{{x + 5}}{{x - 5}} \times \dfrac{{5 - x}}{{{x^2} - 25}}\]. Then we will use the identity \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\] to rewrite the denominator. At last, we will cancel the common terms from the numerator and the denominator and then we will simplify it to find the result.

Complete step by step answer:
We have to simplify \[\dfrac{{x + 5}}{{x - 5}} \div \dfrac{{{x^2} - 25}}{{5 - x}}\].
We will convert the division into multiplication to simplify the given expression. For this, we know that dividing a fraction by another fraction is the same as multiplying the fraction by the reciprocal of the other. So, using this concept in the given expression, we can rewrite the given expression as,
\[ \Rightarrow \dfrac{{x + 5}}{{x - 5}} \times \dfrac{{5 - x}}{{{x^2} - 25}}\]
On rewriting, we get
\[ \Rightarrow \dfrac{{x + 5}}{{x - 5}} \times \dfrac{{ - \left( {x - 5} \right)}}{{{{\left( x \right)}^2} - {{\left( 5 \right)}^2}}}\]
Using the identity \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\]. We get,
\[ \Rightarrow \dfrac{{x + 5}}{{x - 5}} \times \dfrac{{ - \left( {x - 5} \right)}}{{\left( {x - 5} \right)\left( {x + 5} \right)}}\]
Cancelling the common terms from the numerator and the denominator, we get
\[ \Rightarrow - \dfrac{1}{{\left( {x - 5} \right)}} = \dfrac{1}{5-x}\]
Therefore, on simplifying \[\dfrac{{x + 5}}{{x - 5}} \div \dfrac{{{x^2} - 25}}{{5 - x}}\], we get \[ \dfrac{1}{{\left( { 5-x} \right)}}\].

Note:
To solve these types of problems, we should keep in mind the BODMAS rule. This rule is used to remember the order of operations to be followed while solving expressions in mathematics. We have to simplify any expression from left to right by having multiple operators in this order only. Here, B stands for Brackets, O stands for Order of powers or roots, D stands for Division, M stands for Multiplication, A stands for Addition and S stands for Subtraction.