# Find all real numbers a

#### anemone

##### MHB POTW Director
Staff member
Find all real numbers $$\displaystyle a$$ such that the equation $$\displaystyle |x-|x-|x-4||| =a$$ has exactly three real solutions.

#### zzephod

##### Well-known member
Find all real numbers $$\displaystyle a$$ such that the equation $$\displaystyle |x-|x-|x-4||| =a$$ has exactly three real solutions.

The curve

$$\displaystyle y= |x-|x-|x-4|||$$

is piece-wise linear with knots at $$\displaystyle x=4/3,\ 2,\ 4$$ and so considering the shape of the curve $$\displaystyle a=y(2)$$ is the value sought.
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#### MarkFL

$$\displaystyle y= |x-|x-|x-4|||$$
is piece-wise linear with knots at $$\displaystyle x=4/3,\ 2,\ 4$$ and so considering the shape of the curve $$\displaystyle a=y(2)$$ is the value sought.