Is the inequality as follow true?

Let $k > 0, a_i>0$ for $1\le i\le n$ and let $$S:=a_1+a_2+....+a_n$$ Suppose that $$b_i:=S-ka_i \quad\text{ for} \quad 1\le i\le n.$$ Then

$$k(|a_1|+|a_2|+...+|a_n|) \le |b_1|+|b_2|+...+|b_n|+k|S|$$

Equality if only if $S=0$