The maximum number of values of $x$ if $| x - 2 | + | x - 4 | = 2$ is

1

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2

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3

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Infinitely many

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Solution

The correct option is D Infinitely many
$| x - 2 | + | x - 4 | = 2$
Case I:- $x < 2$
$- (x - 2) - (x - 4) = 2$
$\Rightarrow 2 - x + 4 - x = 2$
$\Rightarrow - 2 x = 2 - 6$
$\Rightarrow x = \frac{- 4}{- 2} = 2$
Case II:- $2 \leq x < 4$
$(x - 2) - (x - 4) = 2$
$x - 2 + 4 - x = 2$
$\Rightarrow 2 = 2 \Rightarrow$ For any value of $x$
Case III:- $x \geq 4$
$(x - 2) + (x - 4) = 2$
$\Rightarrow 2 x = 2 + 6$
$\Rightarrow x = \frac{8}{2} = 4$
Hence from the above $3$ case, $x$ can have infinitely many values.

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