A project manager estimates that a project will take $x$hours to complete, where $x > 100$ The goal is for the estimate to be within 10 hours of the time it will actually take to complete the project. If the manager meets the goal and it takes $y$ hours to complete the project, which of the following inequalities represents the relationship between the estimated time and the actual completion time?
$A.\;\;\;\;\;x + y < 10$
$B.\;\;\;\;\;y > x + 10$
$C.\;\;\; - 10 < y - x < 10$
$D.\;\;\; - 10 \leqslant y - x \leqslant 10$

Hint: First assume $x$ as any arbitrary value greater than $100$ because it is given that $x > 100$, and then find the value of $y$, which is the value of the time after 10 hours. Then trying to find the relation between the estimated time and actual time.

Complete step-by-step answer:
It is given that the estimated time that the project will take to complete is $x$ hours, which is greater than $100$. That is, $x > 100$. So, assume the value of $x$ greater than 100.
Firstly, assume that $x = 101$.
Within 10 hours it will reach $y = 111$.
From the above values of $x$ and $y$.
It will give the absolute value of the difference between x and y that is:
$\left| {x - y} \right| < 10$
It can also be expressed as:
$\left| {y - x} \right| < 10$
The above-given relation can also be expressed as:
$ - 10 < y - x < 10$
Thus, the relation between the estimated completion time and the actual completion time is given as:
$ - 10 < y - x < 10$
Therefore, the option$C$ is correct.

Note: The relation between in the estimated time and the actual time is given as:
$\left| {y - x} \right| < 10$
When the value in the absolute value is positive then:
$y - x < 10$
When the value in the absolute value is negative then:
$
   - \left( {y - x} \right) < 10 \\
   \Rightarrow y - x > - 10 \\
 $
Thus, combined the absolute value is expressed as:
$ - 10 < y - x < 10$