Question

How do you simplify $\dfrac{3x-6}{10-5x}$ and find any non-permissible values?

Answer 1

Hint: For simplification, we first need to take $3$ common from the numerator and then $5$ from the denominator. After this, we check the value of x at which the denominator becomes $0$ . Keeping this point (non-permissible) in mind, we then cancel off the $x-2$ terms from the numerator and the denominator.

Complete step by step answer:
The given expression that we have at our disposal in the problem is $\dfrac{3x-6}{10-5x}$ . We need to simplify the given fraction. For this, we need to simplify both the numerator and the denominator of the given fraction. By numerator, we mean the expression in the fraction that lies above the bar and by denominator, we mean the expression of the fraction that lies below the bar.
In the numerator, we can see that the expression is $3x-6$ . We can simplify this expression by using the reverse distributive property. The distributive property is $a\left( b+c \right)=ab+ac$ . By reverse distributive property, we mean $ab+ac=a\left( b+c \right)$ . We can take $3$ common from both the terms \[3x,6\] and get, $3\left( x-2 \right)$ .
Similarly, in the denominator, we can see that the expression is $10-5x$ . We can simplify this expression by using the reverse distributive property. The distributive property is $a\left( b+c \right)=ab+ac$ . By reverse distributive property, we mean $ab+ac=a\left( b+c \right)$ . We can take $5$ common from both the terms \[10,5x\] and get, $5\left( 2-x \right)$ . The fraction becomes,
$\begin{align}
  & \Rightarrow \dfrac{3\left( x-2 \right)}{5\left( 2-x \right)} \\
 & \Rightarrow -\dfrac{3}{5} \\
\end{align}$
But the above simplification is possible only when $x\ne 2$ . This is because if $x=2$ , then, then fraction will become of the form \[\dfrac{0}{0}\] . This is an indeterminate form.

Therefore, we can conclude that the fraction can be simplified to $-\dfrac{3}{5}$ and the non-permissible value is $2$

Note: The simplification in the problem is very easy with simple cancelling off of common terms. But, we should not blindly cancel off the common and should carefully observe at which points the function is not defined. If we don’t observe this, we end up with wrong answers.