Question

The maximum number of values of x if $\left| {x - 2} \right| + \left| {x - 4} \right| = 2$ is
A. 1
B. 2
C. 3
D. Infinitely many

Answer 1

Hint: We can consider 3 cases, when x is less than 2; x is greater than or equal to 2 and less than 4; x is greater than or equal to 4, because we have to consider the interval of real numbers. For each case, find whether the equation is true or not and that will give how many values are possible for x.

Complete step by step solution:
We are given to find the maximum number of values of x if $\left| {x - 2} \right| + \left| {x - 4} \right| = 2$.
So we have to consider the interval of real numbers for which the equation assumes different forms.
Case 1:
When x<2. The first and second terms in $\left| {x - 2} \right| + \left| {x - 4} \right|$ will be negative as x is less than 2.
Therefore, the equation becomes $ - \left( {x - 2} \right) - \left( {x - 4} \right) = 2$
$ \Rightarrow - x + 2 - x + 4 = 2$
$ \Rightarrow - 2x + 6 = 2$
$ \Rightarrow 2x = 4 \Rightarrow x = 2$
Here, we got the value of x as 2 but we considered x value as less than 2. Therefore, x is less than 2 is not a solution.
Case 2:
When $2 \leqslant x < 4$, the first term will be positive (or zero) and second term will be negative in $\left| {x - 2} \right| + \left| {x - 4} \right|$ as x is less than 4 and greater than or equal to 2.
 Therefore, the equation becomes $\left( {x - 2} \right) - \left( {x - 4} \right) = 2$
$ \Rightarrow x - 2 - x + 4 = 2$
$ \Rightarrow 2 = 2$
The above equation is an identity, which means the given equation is always true for the interval $\left[ {2,4} \right)$
Case 3:
When $x \geqslant 4$, $\left[ {4,\infty } \right)$, both the first and second terms will be positive as x is greater than or equal to 4. Second term can also be zero.
Therefore, the equation becomes $\left( {x - 2} \right) + \left( {x - 4} \right) = 2$
$ \Rightarrow x - 2 + x - 4 = 2$
$ \Rightarrow 2x - 6 = 2$
$ \Rightarrow 2x = 8,x = 4$
X is equal to 4 lies in the interval $\left[ {4,\infty } \right)$. This means that x=4 is the solution in this case.
Therefore, in total x=4 and x lies in the interval $\left[ {2,4} \right)$ i.e. x is from the interval $\left[ {2,4} \right]$
Therefore, the no. of integer values in the interval $\left[ {2,4} \right]$ is three i.e. 2, 3 and 4 whereas the no. of real values in this interval is infinite.
Therefore, the maximum number of values of x if $\left| {x - 2} \right| + \left| {x - 4} \right| = 2$ is infinitely many.
So, the correct answer is “Option D”.

Note: Here we are asked to find the maximum no. of values, but clearly they did not mention integer or real. So we have considered real values because integers are also real. All the real values between 2 and 4 satisfy the equation, say x is 3.7 then the equation will be $\left| {3.7 - 2} \right| + \left| {3.7 - 4} \right| = 1.7 + 0.3 = 2$ which satisfies the equation. Anything inside $\left| {} \right|$ results a positive value. This is called absolute value. Absolute value of -2 is 2 and absolute value of 2 is 2.