Question

How do you write an absolute value inequality to represent $\left[ -10,10 \right]$?

Answer 1

Hint: For writing the absolute value inequality corresponding to the given interval, we need to first choose a variable, say x, which will take the values of the given interval $\left[ -10,10 \right]$ so that we can write $x\in \left[ -10,10 \right]$. The two numbers written inside the square bracket to represent an interval means that all of the real numbers lying between those numbers are covered. And since we have square brackets, the end numbers $-10$ and $10$ will also be covered. In other words we can say that x takes all the numbers greater than or equal to $-10$ and less than or equal to $10$ so that we can write the inequalities $x\ge -10$ and $x\le 10$. From these two inequalities, we can write the required absolute value inequality.

Complete step by step solution:
The interval is given in the above question as $\left[ -10,10 \right]$. Let the variable x represent the values inside this interval so that we can write $x\in \left[ -10,10 \right]$. Now, we know that the two numbers written inside the square brackets means that all the numbers between those numbers are there. Also, due to the square brackets, x will take the end numbers $-10$ and $10$ also. Therefore, x takes all the numbers greater than or equal to $-10$ and less than or equal to $10$. Therefore, we can write the below inequalities.
\[\begin{align}
  & \Rightarrow x\ge -10 \\
 & \Rightarrow x\le 10 \\
\end{align}\]
From both of the above inequalities, we can observe that the absolute value of x is less than or equal to $10$. Therefore, the absolute value inequality can be written as
$\Rightarrow \left| x \right|\le 10$

Hence, the absolute value inequality is $\left| x \right|\le 10$.

Note: For solving these types of questions, we must be familiar with the different kinds of representation of the intervals. We must not forget to include the end values of the interval as well, since we have a square bracket, which represents a closed interval. The open interval is represented by circular brackets.