Question

How do you solve \[\left| {8 - 3x} \right| = 8\] ?

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 the mod sign. But the number in the modulus can be positive or negative. So we will assign the modulus to \[8\] as well as \[ - 8\]. 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 \[\left| {8 - 3x} \right| = 8\]
Now we will equate the modulus to \[8\] as well as \[ - 8\].
\[\Rightarrow \left| {8 - 3x} \right| = 8\]
On removing modulus we get,
\[\Rightarrow 8 - 3x = 8\]
Taking 8 on other side we get,
\[\Rightarrow - 3x = 8 - 8\]
On subtracting we get,
\[\Rightarrow - 3x = 0\]
We know that anything multiplied by 0 is always zero.
\[x = 0\]
So the value of x is zero in this case.
Now substituting the modulus to \[ - 8\].
\[\Rightarrow \left| {8 - 3x} \right| = - 8\]
On removing the modulus sign,
\[\Rightarrow 8 - 3x = - 8\]
Transposing 8 on other side,
\[\Rightarrow - 3x = - 8 - 8\]
On taking the mathematical operations we get,
\[\Rightarrow - 3x = - 16\]
Cancelling the minus sign and dividing we get,
\[\Rightarrow x = \dfrac{{16}}{3}\]

Thus the value of x can be \[x = 0\] or \[x = \dfrac{{16}}{3}\]

Note: Here the modulus function has always non negative value as output but the number inside 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.