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

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

Grade:12

4 Answers

Badiuddin askIITians.ismu Expert
147 Points
11 years ago

Dear chilukuri

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

case 1        x>=3

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

or (x-1)+(x-2)+(x-3)>6

x>=4

case 2         2=<x <3

 |x-1|+|x-2|+|x-3|>6

or

(x-1)+(x-2)+(-x+3)>6

x>=6

  so it is not a solution

case 3   1=<x<2

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

or

(x-1)+(-x+2)+(-x+3)>6

x<=-2

 so it is not a solution

case 4  x<1

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

or

(-x+1)+(-x+2)+(-x+3)>6

x<=0  

so final solution

   -∞<x≤0   and   4≤x<


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Badiuddin



Akshay Mr
13 Points
2 years ago
The detailed answer of this question please give me I can't understand the way you Explain the answer
 
Rishi Sharma
askIITians Faculty 646 Points
11 months ago
Dear Student,
Please find below the solution to your problem.

To get rid of the absolute value functions, we need to take 4 cases:
Case 1: x≥3or [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)

Thanks and Regards
Rishi Sharma
askIITians Faculty 646 Points
11 months ago
Dear Student,
Please find below the solution to your problem.

To get rid of the absolute value functions, we need to take 4 cases:
Case 1: x≥3or [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)

Thanks and Regards

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