Real Solutions for Equation |x-|x-|x-4||| = a

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SUMMARY

The discussion focuses on identifying all real numbers \( a \) for which the equation \( |x-|x-|x-4||| = a \) has exactly three real solutions. The participants confirm that the method used to derive the solution is effective and closely resembles the approach taken by the original poster (OP). The equation's structure suggests a piecewise analysis is necessary to determine the conditions under which three solutions occur.

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Find all real numbers $$a$$ such that the equation $$ |x-|x-|x-4||| =a
$$ has exactly three real solutions.
 
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anemone said:
Find all real numbers $$a$$ such that the equation $$ |x-|x-|x-4||| =a
$$ has exactly three real solutions.


The curve

$$y= |x-|x-|x-4||| $$

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

$$y= |x-|x-|x-4||| $$

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

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