Question

How do you solve $\left| { - 5x + \left( { - 2} \right)} \right| = 12$ ?

Answer 1

Hint: We are given simply with an expression having one variable but the variable is in modulus. Modulus is the absolute value or non-negative value of the number in the mod sign. But the number in the modulus can be positive or negative. So we will equate the expression $ - 5x + \left( { - 2} \right)$with both positive as well as negative sign because we know whatever number is present in mod, we have to make it positive. This will give the values of x here.

Complete step-by-step solution:
In the given question, we are required to solve an equation involving modulus function.
So, we have, $\left| { - 5x + \left( { - 2} \right)} \right| = 12$
Opening the modulus function with both positive and negative sign, we get,
$ \Rightarrow \pm \left( { - 5x + \left( { - 2} \right)} \right) = 12$
So, two cases arise in the question.
Either $\left( { - 5x + \left( { - 2} \right)} \right) = 12$ or $ - \left( { - 5x + \left( { - 2} \right)} \right) = 12$
Either $ - 5x = 12 + 2$ or $5x = 12 - 2$
Either $x = - \dfrac{{14}}{5}$ or $x = 2$
So, the solution to the equation involving modulus function given as $\left| { - 5x + \left( { - 2} \right)} \right| = 12$ are: $x = - \dfrac{{14}}{5}$and $x = 2$.

Note: The modulus function always represents the absolute value or non negative value as output but the number inside modulus can be either positive or negative. The definition of the modulus function is:
$\left| x \right| = \left\{
  x\,\,if\,\,x > 0 \\
   - x\,\,if\,\,x < 0 \right\}$
The answer of the equation can also be verified by putting in the value in the equation back.
Apart from this sometimes in mathematics mod also means division and finding the remainder. For example 100 mod 90 is 10. Such that 100 is divided by 90 to give 10 as remainder. But the modulus in our problem above is the symbol of two vertical parallel lines that mean absolute value.