Question

How do you solve \[\left| {2x + 6} \right| - 4 = 20\] ?

Answer 1

Hint: We are given simply with an expression having one variable but the variable is in modulus. Modulus is the absolute or non-negative value of the number in mod sign. But the number in the modulus can be positive or negative. So we will assign the modulus to \[24\] as well as to \[ - 24\]. Because we know whatever number is present in mod is always output as positive. This will give the value of x here.

Complete step by step solution:
Given that \[\left| {2x + 6} \right| - 4 = 20\]
Now we will shift 4 to other side of equation,
\[\left| {2x + 6} \right| = 20 + 4\]
\[\left| {2x + 6} \right| = 24\]
So let’s consider two equations,
\[ \left| {2x + 6} \right| = 24\] and \[ \left| {2x + 6} \right| = - 24\].
Now we will solve them.
\[\left| {2x + 6} \right| = 24\]
\[2x + 6 = 24\]
Taking 6 on RHS we get,
\[2x = 24 - 6\]
On solving we get,
\[2x = 18\]
Dividing 18 by 2 we get,
\[\Rightarrow x = \dfrac{{18}}{2} = 9\]
This is the solution. Now for second equation,
\[\left| {2x + 6} \right| = - 24\]
\[2x + 6 = - 24\]
Taking 6 on RHS we get,
\[2x = - 24 - 6\]
On solving we get,
\[2x = - 30\]
Dividing 30 by 2 we get,
\[\Rightarrow x = \dfrac{{ - 30}}{2} = - 15\]
This is our solution.
So values of x for the given expression are \[ + 9\] and \[ - 15\].
So, the correct answer is “ \[ x = + 9\] and \[ x = - 15\]”.

Note: Here the modulus function has always non-negative value as output but the number inside the modulus can be either positive or negative.
\[\left| x \right| = \left\{ {\begin{array}{*{20}{c}}
  {x,x \geqslant 0} \\
  { - x,x < 0}
\end{array}} \right.\]
Apart from this sometimes in mathematics modulus 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.