Question

How do you solve | 5 – x | = - | x – 5|?

Answer 1

Hint: We will break the question in two cases, when x is greater than 5 or when x is less than equal to 5. Then, we will define both the modulus functions for that condition and solve it.

Complete step by step solution:
We are given that we are required to solve | 5 – x | = - | x – 5|.
We will first break the situation in two different cases:-
Case 1: When $x > 5$
By using the definition of modulus function, we will then obtain the following equation with us:-
\[ \Rightarrow \] | 5 – x| = - (5 – x) = x – 5
\[ \Rightarrow \] | x – 5| = x – 5
Substituting both of these in the given equation | 5 – x | = - | x – 5|, we will then obtain the following equation with us:-
\[ \Rightarrow \] x – 5 = - (x – 5)
Simplifying the right hand side of the above equation, we will then obtain the following equation with us:-
\[ \Rightarrow \] x – 5 = 5 – x
Taking x from subtraction in the right hand side to addition in the left hand side and taking 5 from subtraction in the left hand side to addition in the right hand side, we will then obtain the following equation:-
\[ \Rightarrow \] 2x = 10
Thus, x = 5.
Case 2: When $x \leqslant 5$
By using the definition of modulus function, we will then obtain the following equation with us:-
\[ \Rightarrow \] | 5 – x| = 5 – x
\[ \Rightarrow \] | x – 5| = - (x – 5) = 5 – x
Substituting both of these in the given equation | 5 – x | = - | x – 5|, we will then obtain the following equation with us:-
\[ \Rightarrow \] 5 – x = - (5 – x)
Simplifying the right hand side of the above equation, we will then obtain the following equation with us:-
\[ \Rightarrow \] 5 – x = x - 5
Taking x from subtraction in the left hand side to addition in the right hand side and taking 5 from subtraction in the right hand side to addition in the left hand side, we will then obtain the following equation:-
\[ \Rightarrow \] 2x = 10
Thus, x = 5.
Hence, the required answer is x = 5.

Note:
The students must also note that they may also verify it by putting in x = 5 in the given equation.
Now, we will get 0 on both the sides of the equation, therefore, the answer x = 5 has been verified.
The students must also note that the definition of modulus function which has been used in the above solution is given as follows:-
$ \Rightarrow |x| = \left\{ {\begin{array}{*{20}{c}}
  {x,x > 0} \\
  { - x,x < 0}
\end{array}} \right.$