Find the values of x for which $| x - 2 | + | x - 8 | = 10$ .

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Solution

Given: $| x - 2 | + | x - 8 | = 10$

Case 1: For $x < 2$

$- (x - 2) - (x - 8) = 10$

$- x + 2 - x + 8 = 10$

$- 2 x + 10 = 10$

$- 2 x = 0$

$\Rightarrow$ $x = 0$

Case 2: For $2 \leq x < 8$

$(x - 2) - (x - 8) = 10$

$x - 2 - x + 8 = 10$

$6 = 10$ No solution

Case 3: For $x \geq 8$

$x - 2 + x - 8 = 10$

$2 x - 10 = 10$

$2 x = 20$

$\Rightarrow$ $x = 10$

Hence, the values of $x$ are 0 and 10.

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