MHB What are the Values of k that Satisfy the Given Equation?

  • Thread starter Thread starter anemone
  • Start date Start date
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Find all numbers of $k$ such that the equation $|x+|x|+k|+|x-|x|-k|=2$ has exactly three solutions.
 
Mathematics news on Phys.org
Hint:

What would happen to the given equation when we replace $x$ by $-x$?
 
k=-1 is the only one, 2 is the result for x={-1,0,1}
 
Thanks, RLBrown for giving us the correct end result for this challenge!:)

Solution of other:

If $x$ is a solution to the given equation, then $-x$ is also a solution, since

$|(-x)+|-x|+k|+|(-x)-|-x|-k|=2$

$|-x+|x|+k|+|-[x+|x|+k]|=2$

$|-[x-|x|-k]|+|x+|x|+k|=2$

$|x-|x|-k|+|x+|x|+k|=2$ which is the same as the original equation $|x+|x|+k|+|x-|x|-k|=2$.

Thus, in order for the equation to have an odd number of solutions, it is clear that $x=0$ must be one of these.

Substituting $x=0$ into the equation, we see that

$|k|+|-k|=2$

$|k|+|k|=2$

$2|k|=2$

$k|=\pm 1$

[TABLE="class: grid, width: 680"]
[TR]
[TD]If $k=1$ and $x\ge 0$, then the equation becomes

$|2x+1|+1=2$ and so $x=0$

But $-x$ is a solution iff $x$ is, so there is only one solution in this case.[/TD]
[TD]If $k=-1$ and $x\ge 0$, then the equation becomes

$|2x-1|+1=2$ and so $x=0,\,1$

Thus, in this case, we have three solutions where $x=-1,\,0,\,1$

[/TD]
[/TR]
[/TABLE]

$\therefore k=-1$ is the unique number for which the equation has exactly three solutions.
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Back
Top