Minimum Value

Algebra Level 3

Find the minimum value of $|x-2| + |x-4| + |x-5| + |x-7|.$

Notation : $| \cdot |$ denotes the absolute value function .

2 solutions

Rishabh Jain
Jul 14, 2016

Write the expression as:

$\mathcal G=|x-2|+|x-4|+|5-x|+|7-x|$

Applying general form of triangle inequality:

$\mathcal G\ge |(x-2)+(x-4)+(5-x)+(7-x)|=\boxed 6$

Equality holds when $4\le x\le 5$ .

See here for a proof of generalised form of triangle inequality.

Naitik Sanghavi
Jul 14, 2016

Minimum value of this will occur at x=4.5=(2+4+5+7)/4.