Question

If x is a real number and $\left| x \right|<5$ , then
A. $x\ge 5$
B. $-5< x< 5$
C. $x\le -5$
D. $-5\le x\le 5$

Answer 1

Here we have been given a modulus of a real number which is less than a certain value. We have to find a simple way to write the value or we can say the range of the number when the modulus sign is removed. Firstly we write the basic way a modulus is defined and by using it we will get the two values of the variable. Finally we will combine the two values and get the desired answer.

Complete step-by-step solution:
The real number $x$ is given as follows:
$\left| x \right|<5$…..$\left( 1 \right)$
Now as we know that modulus of a real number $x$ is denoted by $\left| x \right|$ and is defined as,
\left| x \right|=\left\\{ \begin{aligned} & x; \text{if}\text{ }x\ge 0 \\\ & -x ; \text{if}\text{ }x< 0 \\\ \end{aligned} \right\\}
So we can write the value in equation (1) as follows:
Firstly if,
$x< 5$ If $x\ge 0$
This means we get,
$0\le x<5$…..$\left( 2 \right)$
Secondly if,
$x>-5$ If $x<0$
This means we get,
$-5< x< 0$……$\left( 3 \right)$
On combining equation (2) and (3) we get the below value,
$-5< x< 5$
The above inequality can be shown graphically as:

Hence the correct option is (B).

Note: Modulus of Real Number is also known as absolute value or numerical value where $x$ is denoted by $\left| x \right|$ . It is the distance that goes from the number to its origin which makes part of the real straight line. It produces the magnitude of the number of variables. The value that comes out from a modulus is always positive no matter what is the sign of the number inside the modulus. If the variable sign is positive then we get only output but if the variable sign is negative we have to split the output into two and hence we get two values.