Question

How do you solve $\left| x \right| = 6?$

Answer 1

Hint:The given function is a modulus function, and modulus function works according to its argument or input (value of independent variable, generally denoted as “x”).The modulus function opens according to its input values as follows: It will have positive sign when input of modulus is greater than or equals to zero, i.e. $\left| x \right| = x,\;{\text{if}}\;x \geqslant 0$
And it will open with negative sign when its input is less than zero or it has negative input, i.e. $\left| x \right| = - x,\;{\text{if}}\;x < 0$
In case we do not know about the input of the modulus function, we open it with both possibilities.

Complete step by step answer:
In order to solve the given equation $\left| x \right| = 6$, we have to solve this in two cases, first when the argument of the modulus function has domain less than zero and second when the argument of the modulus function has domain greater than or equal to zero.Argument of the given modulus function is $x$.

Case I: When $x < 0$
In this case modulus will open with negative sign,
$\left| x \right| = 6 \\
\Rightarrow - x = 6 \\
\Rightarrow x = - 6 \\ $
Case II: When $x \geqslant 0$
In this case modulus will open with positive sign,
$\left| x \right| = 6 \\
\Rightarrow x = 6 \\ $
Taking the solutions from both cases, the required solution will be $x = \;\{ - 6,\;6\} $

Note:Since this is a very basic question including modulus function, there are functions which consist of complex modulus functions, like modulus in modulus, for that type of questions you have to make sub cases of cases. And also check the answer from each case for the original equation and the domain taken for the case.