Question

Chap: Linear inequations

solve: |x-1|+|x+2|≥4​

Answer 1

Answer:

Think of |a−b| as the ‘‘ distance ” between the points representing the real numbers a and b . So |x−1| is the distance between x and 1 , and |x−2| is the distance between x and 2 . Therefore, if x lies between 1 and 2 , |x−1|+|x−2| is just the distance between 1 and 2 . This distance is 1 .

If you go either side of these two points 1 , 2 , by a distance d , say, the sum of the distances, given by |x−1|+|x−2| , increases by 2d . For x lying between 1 and 2 , the sum of distances equals 1 ; so we need to consider points that increase this sum by at least 3 . For this, we must move at least 32 units on either side of the two points 1 and 2: so x≤1−32=−12 or x≥2+32=72 .

Therefore, |x−1|+|x−2|≥4⇔x∈(−∞,−12]∪[72,∞) . ■

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