understanding absolute value with inequality

Question

$|x|\le a$
when $a>0$
then $-a\le x \le a$
but and this is where I don't understand:
when $a < 0$
there is no solution.

Why this is logical? please explain in detail as much as possible.

Answer 1

The number $|x|$ will always be nonnegative. That is, for every value of $x$, the inequality $|x|\geq 0$ will be true, and since $0>a$ is also true, from these two inequalities, we can conclude that $|x| > a$, which means $|x|\leq a$ is impossible.

Answer 2

@Matti.P explained it pretty well in the comment section - $|x|$ is always non-negative, hence the statement $|x|\leq a $ for $a<0$ can't have any solutions.

If you want to visualise things, try to draw the function $f(x)=|x|$. It should have this v-form since $f(x)=x$ for $x\geq0$ and $f(x)=-x$ for $x<0$ as the $|x|=x$ for $x\geq0$ and $|x|=-x$ for $x<0$.

What you should do next is try to draw the graph of the function $g(x)=a$ for a constant $a$. It is a line parallel to the y-axis. You will then notice that if $a<0$ then the lines never meet, if $a=0$ they meet at $0$ and if $a>0$ then they meet at $2$ points - $a$ and $(-a)$ respectively.