Question

How do you solve |8 – 2x| = 4?

Answer 1

Hint: We will first write that quantity inside the modulus is already positive and then solve the equation generally, then we will take in account the possibility that the quantity inside the modulus is negative and get the required answer.

Complete step-by-step answer:
We are given that we are required to solve |8 – 2x| = 4.
Using the definition of the modulus function, we have:-
Either 8 – 2x = 4 …………….(1) or – (8 – 2x) = 4 ……………..(2)
We will solve both the equations 1 and 2 one by one now, to find the possible answers.
Considering the equation 1: 8 – 2x = 4
Taking 8 from addition in the left hand side to subtraction in right hand side, we will then obtain the following equation:-
$ \Rightarrow $- 2x = 4 – 8
Simplifying the calculations in the right hand side of the above equation, we will then obtain the following equation:-
$ \Rightarrow $- 2x = – 4
Dividing both sides of the equation given above by – 2, we will then obtain the following equation:-
$ \Rightarrow $x = 2
Now, we will solve the second equation.
Considering the equation 2: - (8 – 2x) = 4
Simplifying the left hand side of the above equation, we will then obtain the following equation:-
$ \Rightarrow $- 8 + 2x = 4
Taking 8 from subtraction in the left hand side to addition in right hand side, we will then obtain the following equation:-
$ \Rightarrow $2x = 4 + 8
Simplifying the calculations in the right hand side of the above equation, we will then obtain the following equation:-
$ \Rightarrow $2x = 12
Dividing both sides of the equation given above by 2, we will then obtain the following equation:-
$ \Rightarrow $x = 6

Therefore, the possible values of x are 2 and 6.

Note:
The students must know the definition of modulus function which we used in the beginning of the solution.
If we have a function f (x) = |x| which is known as modulus function, then we have:-
$ \Rightarrow f(x) = \left\{ {\begin{array}{*{20}{c}}
  {x,x \geqslant 0} \\
  { - x,x \leqslant 0}
\end{array}} \right.$
We used the same definition by just replacing x by 8 – 2x, thus we got:-
$ \Rightarrow $|8 – 2x| = 8 – 2x, whenever $8 - 2x \geqslant 0$
$ \Rightarrow $|8 – 2x| = 8 – 2x, whenever $ - 2x \geqslant - 8$
$ \Rightarrow $|8 – 2x| = 8 – 2x, whenever $x \leqslant 4$
$ \Rightarrow $|8 – 2x| = - (8 – 2x), whenever $8 - 2x \leqslant 0$
$ \Rightarrow $|8 – 2x| = 2x - 8, whenever $ - 2x \leqslant - 8$
$ \Rightarrow $|8 – 2x| = 2x - 8, whenever $x \geqslant 4$
Thus, we got: $|8 - 2x| = \left\{ {\begin{array}{*{20}{c}}
  {8 - 2x,x \leqslant 4} \\
  {2x - 8,x \geqslant 4}
\end{array}} \right.$