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Find all real numbers a

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,713
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
Feb 3, 2013
134
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.
.
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
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.
.
The OP is having trouble logging on to MHB, and has asked me to congratulate you on her behalf on finding the correct answer, and says the method you have used is nearly identical to the method she used in finding the solution.