Question

Solve |x−1|+|x−2|+|x−3|≥6|x−1|+|x−2|+|x−3|≥6​

Answer 1

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8th

Maths

Linear Equations in One Variable

Reducing Equations to Linear Form

If x satisfies | x - 1 |...

MATHS

If x satisfies ∣x−1∣+∣x−2∣+∣x−3∣≥6 then prove that all numbers x which satisfy the above relation are given by x≤0 or x≥4

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ANSWER

Here the change point are x = 1 , 2 , 3

Hence we consider the following case

(I) x < 1

(II) 1 < x < 2

(III) 2 < x < 3

(IV) x > 3

case (I) x < 1

-(x - 1) - ( x - 2) - (x - 3) ≥ 3

-3x + 6 ≥6 or -3x ≥ 0 ∴ x ≥\, 0

Which is < 1 and hence the solution

case (II) 2 ≥ x <3

(x - 1) - ( x - 2) - (x - 3) ≥ 3

-3x + 6 ≥6 or -x ≥ 2 x ≥\, -2

This does not satisfy given condition of case (II) Hence no solution

case (III) 2 ≥ x <3

(x - 1) - ( x - 2) - (x - 3) ≥ 3

x ≥\, 6

This does not satisfy given condition of case (III) Hence no solution

case (IV) x ≥3

(x - 1) - ( x - 2) - (x - 3) ≥ 3

x ≥\, 14 or x ≥\, 4

This does not satisfy given condition of case (III) Hence no solution

Thus the required solution by case I are IV are x ≥\, 0 or x ≥\, 4

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