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

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The discussion centers on finding real numbers \( a \) for which the equation \( |x-|x-|x-4||| = a \) yields exactly three real solutions. Participants explore the behavior of the nested absolute value function and its critical points to determine the values of \( a \). The conversation highlights the importance of understanding the structure of the equation to identify when it intersects with horizontal lines representing \( a \). Additionally, there is acknowledgment of a member's successful method in solving the equation, indicating a shared approach among participants. The focus remains on the mathematical analysis required to solve the equation effectively.
anemone
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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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