What does m<|x| imply

absolute valuealgebra-precalculusfunctions

I know $|x|\le M \ \implies -M\le x \le M$ but what does $m\le|x|$ imply?

Can I just plainly interpret this as $m\le x$?

and if I say $m\le x \le M$ does this still mean $-M \le x \le M$?

Best Answer

I am assuming $m > 0$. $|x| \ge m$ implies $x \ge m$ or $x \le - m$. To see this, I recommend you consider $m = 2$ and look at the following graph. The red curve is $y = |x|$ and the blue horizontal line is $y = 2$, positioned $2$ units above the $X$-axis.

enter image description here

If $|x| \le 2$, you're looking at the region (line segment) on the $X$-axis which is contained between the points $-2$ and $2$. This is clear from the picture. If $|x| \ge 2$, you're looking at two rays, one going to $\infty$ and another to $-\infty$, emanating from $2$ and $-2$ respectively. Hope this helps.

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