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Problem:

Prove that : if ,for all,x $ a|x|+bx\geq 0$ then $ a\geq |b|$

Proof:

Suppose that for each x we have $a|x|+bx\geq 0$...................................................................1

we must show that : $a\geq |b|$

Because (1) holds for each x it will hold and for x=1 and x=-1.

From (1) for x=1 we have that $a+b\geq 0$ or $a\geq -b$,while for x=-1 that $a-b\geq 0$ or $a\geq b$.

Hence we have $a\geq -b$ and $a\geq b$ or $a\geq |b|$.

Is that proof correct??