Question

Solve each of equations in R :

|x - 1| + |x - 2| + |x - 3| > 6​

Answer 1

Answer:

Step-by-step explanation:

|x−1|+|x−2|+|x−3|≥6

To get rid of the absolute value functions, we need to take 4 cases:

Case 1: x≥3 or [3,∞)

Here all the bars will open with a positive sign.

x−1+x−2+x−3≥6

3x≥12

x≥4

Case 2: 2≤x<3 or [2,3)

Here, only |x−3| will open with a negative sign.

x−1+x−2−x+3≥6

x≥6

But this lies out of [2,3)

So no solution

Case 3: 1≤x<2 or [1,2)

Here only |x−1| will open with a positive sign.

x−1−x+2−x+3≥6

−x≥2

x≤−2

Case 4: x<1(−∞,1)

Here all bars will open with a negative sign.

−x+1−x+2−x+3≥6

x≤0

Eliminating the redundant solutions, we get the final answer as:

x∈(−∞,0]∪[4,∞)

or  x∈R−(0,4)