I have seen the teacher solving this equation, but I cannot understand the solution. Would someone please explain a solution to this equation?
$|x|+x=0$
Solution:
$x+(-x)=0$
$0=0. $
I have seen the teacher solving this equation, but I cannot understand the solution. Would someone please explain a solution to this equation?
$|x|+x=0$
Solution:
$x+(-x)=0$
$0=0. $
You have the equation $|x|+x=0$. When you see the absolute value, you know that there are two cases to consider, $x\ge 0$ and $x<0$. Let’s look first for a solution with $x\ge 0$.
If $x\ge 0$, then $|x|=x$, and the equation becomes $x+x=0$, or $2x=0$; this has $x=0$ as its only solution.
If $x<0$, then $|x|=-x$, and the equation becomes $-x+x=0$, or $0=0$. This is always true, so every $x<0$ is a solution. Take $x=-3$; then $|x|=|-3|=3$, and $|x|+x=3+(-3)=0$. The same sort of thing happens with every negative number.
The solutions to the equation $|x|+x=0$ are therefore those $x$ such that $x\le 0$; in interval notation that’s the set $(\leftarrow,0]$ or $(-\infty,0]$. In set-builder notation it’s $\{x:x\le 0\}$.
Well, if $x\geq0$ then $|x|=x$ hence the equation is $x+x=0$ thus $x=0$.
If $x<0$ then $|x|=-x$ so you have $(-x)+x=0$ thus every negative $x$ is also a solution.