What is |x-1|+|x-2|+|x-3|>= 6 ?

The solution to |x-1|+|x-2|+|x-3|>= 6 is x = 4

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|x-1|+|x-2|+|x-3|>= 6

∣ x - 1 ∣ + ∣ x - 2 ∣ + ∣ x - 3 ∣ \geq 6

x \leq 0 or x \geq 4

Interval Notation

(- \infty, 0] \cup [4, \infty)

Solution steps

∣ x - 1 ∣ + ∣ x - 2 ∣ + ∣ x - 3 ∣ \geq 6

Find positive and negative intervals

x < 1, 1 \leq x < 2, 2 \leq x < 3, x \geq 3

Solve the inequality for each interval

x \leq 0 or No Solution or x \geq 4

Merge Overlapping Intervals

x \leq 0 or x \geq 4

\frac{( ∣ x ∣ + 4 )}{( ∣ x ∣ + 3 )} \geq 0

∣ x - 1 ∣ < ∣ x ∣

x^{2} + x + ∣ x ∣ + 1 < 0

∣ x + 1 ∣ + ∣ x ∣ > 3

x^{2} + x + ∣ x ∣ + 1 \leq 0

What is |x-1|+|x-2|+|x-3|>= 6 ?
The solution to |x-1|+|x-2|+|x-3|>= 6 is x<= 0\lor x>= 4