For what range of a-values does $||x|-2|=a$ have only 2 possible answers? (without drawing a graph)

Question

How can i solve "For what range of a-values does the equation $||x|-2|=a$ have only 2 possible answers?" Without using graphs

by using a graph I know the answer is a>2 or {0} as in the picture :

Answer 1

If $a<0$ then as $||x|-2|>0$ we get $0$ solutions

If $a \geq 0$ then we get $2$ cases:

$|x|-2=a$ or $|x|-2=-a$ which is equivalent to

$|x|=a+2$ or $|x|=2-a$

If $2 > a \geq -2$ then both $a+2$ and $2-a$ are $\geq 0$, hence we'd get $4$ solutions - $2$ in each of the above $2$ cases if $2-a \neq a+2$, i.e. $a \neq 0$. In this special case, we get $2$ solutions

If $-2 > a$ then both $a+2$ and $2-a$ are $< 0$, hence no solutions

If $a>2$, then $2-a <0 < a+2$, hence $2$ solutions

As you can see, it's exactly the same as plotting the equation, but with more casework

Answer 2

By resolving the outer absolute value, either $$ |x|=2+a $$ or $$ |x|=2-a. $$ Now you need the value of $a$ so that one of these cases has no solutions.

Answer 3

• If $x<-2$, then: $$||x|-2| = |-x-2| = -x-2= a$$

• If $-2\leq x <0$, then: $$||x|-2|=|-x-2|=2+ x = a$$

• If $0\leq x<2$, then: $$||x|-2|=|x-2|=2-x= a$$

• If $x \geq 2$, then: $$||x|-2|=|x-2|=x-2=a$$

Answer 4

I think, you picture is very clear. Only for $a>2$ and for $a=0$ we have only 2 solutions. For $a=2$ we have exactly 3 solutions. For $0<a<2$ we have 4 solutions. For $a<0$ we not have any solution. Let us analyze each case.
$a>2$ : $$ ||x|-2|>2 \\ $$ We have $|x|-2>2$ and $|x|-2<-2$. But in this second case, $|x|-2<-2$ this will give us $|x|<0$ which is not possible. So for $a>2$ we only have 2 solutions using $|x|-2>2$ : $$ |x|-2>2 \\ |x|>4 $$ This give x>4 and x<-4, 2 solutions.
Let us show now $a=0$ : $$ ||x|-2|=0 \\ |x|-2=0 $$ So , for a=0, we have only 2 solutions : $x=2$ and $x=-2$
For $a=2$ we will have : $$ ||x|-2|=2 \\ $$ We have $|x|-2=2$ and $|x|-2=-2$. The first equation give us 2 solution : x=4 and x=-4. And this second equation give us x=0.
For $0<a<2$ we have : $$ 0<||x|-2|<2 $$ We have $0<|x|-2<2$ and $-2<|x|-2<0$ Thiis first equation give us for x, $2<|x|<4$, so $2<x<4$ and $-4<x<-2$. For this second equation $-2<|x|-2<0$, we get $0<|x|<2$, so $0<x<2$ and $-2<x<0$.
Greatings. Daniel