Question

Fie x,y numere reale. Aratati ca : a) |x-1|+|x-2|+|x-3|>=|x|

Answer 1

Explicație pas cu pas:

Aplicam |x|+|y|>=|x+y|

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

Stim ca |x|=|-x| deci putem spune ca |x-3|=|3-x|

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

dar |2x-3|+|3-x|>=|2x-3+3-x| => |2x-3|+|3-x|>=|x|

deci |x-1|+|x-2|+|3-x|>=|x| <=> |x-1|+|x-2|+|3-x|>=|x|