Question

How do you find the distance between $9$ and $– 12$ using absolute values?

Answer 1

Hint: We will first write them in the format of absolute values and then add the required quantities inside and then look if it is positive or negative to get the positive result.

Complete step-by-step solution:
We know that absolute distance between two real numbers $x$ and $y$ is given by $x – y$.
Now, since we need to find the absolute distance between two real numbers, therefore the distance will be $| x – y|$.
We are given that we are required to find the absolute distance between $9$ and $– 12$.
So, let us assume that $x = 9$ and $y = - 12$.
$ \Rightarrow $$| x – y| = | 9 – ( - 12) |$
Since we know that when we have two negatives, we get positive. Therefore, we will get the following equation:-
$ \Rightarrow $$| x – y| = | 9 + 12 |$
Let us add the given values in the right hand side of the above equation, we will then get the following equation:-
$ \Rightarrow $$| x – y| = | 21 |$
Since $21$ is already positive, we would not have to change any of its sign and we will get the following equation:-
$ \Rightarrow $$| x – y| = 21$

Hence, the absolute distance between 9 and – 12 is 21 units.

Note: The students must note that the absolute function is given as following:-
$ \Rightarrow |x| = \left\{ {\begin{array}{*{20}{c}}
  {x,x > 0} \\
  {0,x = 0} \\
  { - x,x < 0}
\end{array}} \right.$
Therefore, whatever be the value of $x$, whenever we take it out of the modulus sign, we always get a positive result.
The students must note that even if they would have taken $x$ to be $– 12$ and $y$ to be $9$, even then we would have got the same result. See the following for the same:-
We are given that we are required to find the absolute distance between $9$ and $– 12$.
So, let us assume that $y = 9$ and $x = - 12$.
$ \Rightarrow $$| x – y| = | - 12 – 9 |$
Let us add the given values in the right hand side of the above equation, we will then get the following equation:-
$ \Rightarrow $$| x – y| = | - 21 |$
Since $21$ is negative, we would have to change its sign and we will get the following equation:-
$ \Rightarrow $$| x – y| = 21$
So, the absolute difference is $21$ units.