What are the numbers x, for which |x – 2| + |x – 8 | = 6?

To find the value of x we will take different cases.

Case1 : when x>2, |x -2| = x-2 and x>8, |x – 8| = x-8

x-2 + x-8 = 6

⇒ 2x-10 = 6

⇒ 2x = 6 + 10

⇒ 2x = 16

⇒ x = 8 ……….eq(1)

Case2 : when x>2, |x -2| = x-2 and x<8,|x – 8| = -(x-8)

(x-2) + (-x + 8) = 6

No solution as x gets cancelled out………..eq(2)

Case3 : when x<2, |x -2| = -(x-2) and x>8,|x – 8| = x-8

⇒ -x + 2 + x-8 = 6

No solution as x gets cancelled out ………………..eq(3)

Case4 : when x<2, |x -2| = -(x-2) and x<8,|x – 8| = -(x-8)

(-x + 2) + (-x + 8) = 6

⇒ -2x + 10 = 6

⇒ -2x = 6-10

⇒ -2x = -4 ⇒ x = 2 …………………..eq(4)

Now from eq(1) ans eq(4), we have x = 8 and x = 2 as the solution of the equation.