Question

How do you simplify the rational expressions and state restrictions on the variable for \[\dfrac{{3x + 6}}{{5x + 10}}\]?

Answer 1

Hint:The given question describes the operation of addition/subtraction/ multiplication/division. To solve this problem we need to know how to find the greatest common factor between the numerator and denominator. If the denominator value is zero the whole value becomes infinity. By using this above hint we would state the restrictions.

Complete answer:
The given question is given below,
\[\dfrac{{3x + 6}}{{5x + 10}} = ?\]
In the first step, we want to simplify the rational expressions given in the question. We would simplify the numerator. Let’s see the numerator, we can find the term \[3\] is common in both \[x\] terms and a constant term. So, the numerator term becomes,
\[3\left( {x + 2} \right)\]
Next, we would simplify the denominator. Here the term \[5\] is common in the \[x\]term and constant term. So, the denominator term becomes,
\[5\left( {x + 2} \right)\]
So, the given question can be written as given below,
\[
  \dfrac{{3x + 6}}{{5x + 10}} = \dfrac{{3\left( {x + 2} \right)}}{{5\left( {x + 2} \right)}} \\
  \dfrac{{3x + 6}}{{5x + 10}} = \dfrac{3}{5} \to \left( 1 \right) \\
 \]
As a next step, we would state the restrictions on the variable. The given equation is,
\[\dfrac{{3x + 6}}{{5x + 10}}\]
The main restriction is the denominator would not be equal to zero. If we substitute \[x = - 2\], the denominator value becomes zero. So, the value of \[x\] would not be equal to \[ - 2\].
Restriction: \[x \ne - 2\]

Note: To solve this type of question we would use the operation of addition/subtraction/multiplication/division. If the same term is present in the numerator and denominator, we can cancel the term. Note that the denominator value would not be equal to zero. So, we create a condition, which is x is not equal to -2(that is \[x \ne - 2\]).