Solving advanced complex inequalities












-1












$begingroup$


$|z − 2| + |z + 2| ≤ 2$



I've tried solving it like this:



$|x+iy - 2| + |x+iy + 2| le 2\
|x - 2 + iy| + |x + 2 + iy| le 2\
sqrt{(x-2)^2+(y)^2} + sqrt{(x+2)^2+(y)^2} le 2\
sqrt{(x-2)^2+(y)^2} le 2 - sqrt{(x+2)^2+(y)^2}\
sqrt{x^2 - 4x + 4 + y^2} le 2 - sqrt{x^2 + 4x + 4 + y^2}\
x^2 - 4x +4 + y^2 le 4 - 4sqrt{x^2 + 4x + 4 + y^2} + x^2 + 4x + 4 + y^2\
-8x le 4 - 4sqrt{x^2 + 4x + 4 + y^2}\
-2x le 1 - sqrt{x^2 + 4x + 4 + y^2}\
-2x - 1lesqrt{x^2 + 4x + 4 + y^2}\
4x^2+4x + 1 le x^2 + 4x + 4 + y^2\
3x^2-y^2 le 3$



Which seems to be incorrect, what am I doing incorrectly in my procedure? Is my logic correct?










share|cite|improve this question









$endgroup$












  • $begingroup$
    I can see one wrong step here: From $-2x-1leq sqrt{x^2+4x+4+y^2}$ we cannot deduce the inequality for their squares, that you have written. An example: $-10 leq 3$ but $100notleq 9.$
    $endgroup$
    – user376343
    Dec 5 '18 at 11:37
















-1












$begingroup$


$|z − 2| + |z + 2| ≤ 2$



I've tried solving it like this:



$|x+iy - 2| + |x+iy + 2| le 2\
|x - 2 + iy| + |x + 2 + iy| le 2\
sqrt{(x-2)^2+(y)^2} + sqrt{(x+2)^2+(y)^2} le 2\
sqrt{(x-2)^2+(y)^2} le 2 - sqrt{(x+2)^2+(y)^2}\
sqrt{x^2 - 4x + 4 + y^2} le 2 - sqrt{x^2 + 4x + 4 + y^2}\
x^2 - 4x +4 + y^2 le 4 - 4sqrt{x^2 + 4x + 4 + y^2} + x^2 + 4x + 4 + y^2\
-8x le 4 - 4sqrt{x^2 + 4x + 4 + y^2}\
-2x le 1 - sqrt{x^2 + 4x + 4 + y^2}\
-2x - 1lesqrt{x^2 + 4x + 4 + y^2}\
4x^2+4x + 1 le x^2 + 4x + 4 + y^2\
3x^2-y^2 le 3$



Which seems to be incorrect, what am I doing incorrectly in my procedure? Is my logic correct?










share|cite|improve this question









$endgroup$












  • $begingroup$
    I can see one wrong step here: From $-2x-1leq sqrt{x^2+4x+4+y^2}$ we cannot deduce the inequality for their squares, that you have written. An example: $-10 leq 3$ but $100notleq 9.$
    $endgroup$
    – user376343
    Dec 5 '18 at 11:37














-1












-1








-1





$begingroup$


$|z − 2| + |z + 2| ≤ 2$



I've tried solving it like this:



$|x+iy - 2| + |x+iy + 2| le 2\
|x - 2 + iy| + |x + 2 + iy| le 2\
sqrt{(x-2)^2+(y)^2} + sqrt{(x+2)^2+(y)^2} le 2\
sqrt{(x-2)^2+(y)^2} le 2 - sqrt{(x+2)^2+(y)^2}\
sqrt{x^2 - 4x + 4 + y^2} le 2 - sqrt{x^2 + 4x + 4 + y^2}\
x^2 - 4x +4 + y^2 le 4 - 4sqrt{x^2 + 4x + 4 + y^2} + x^2 + 4x + 4 + y^2\
-8x le 4 - 4sqrt{x^2 + 4x + 4 + y^2}\
-2x le 1 - sqrt{x^2 + 4x + 4 + y^2}\
-2x - 1lesqrt{x^2 + 4x + 4 + y^2}\
4x^2+4x + 1 le x^2 + 4x + 4 + y^2\
3x^2-y^2 le 3$



Which seems to be incorrect, what am I doing incorrectly in my procedure? Is my logic correct?










share|cite|improve this question









$endgroup$




$|z − 2| + |z + 2| ≤ 2$



I've tried solving it like this:



$|x+iy - 2| + |x+iy + 2| le 2\
|x - 2 + iy| + |x + 2 + iy| le 2\
sqrt{(x-2)^2+(y)^2} + sqrt{(x+2)^2+(y)^2} le 2\
sqrt{(x-2)^2+(y)^2} le 2 - sqrt{(x+2)^2+(y)^2}\
sqrt{x^2 - 4x + 4 + y^2} le 2 - sqrt{x^2 + 4x + 4 + y^2}\
x^2 - 4x +4 + y^2 le 4 - 4sqrt{x^2 + 4x + 4 + y^2} + x^2 + 4x + 4 + y^2\
-8x le 4 - 4sqrt{x^2 + 4x + 4 + y^2}\
-2x le 1 - sqrt{x^2 + 4x + 4 + y^2}\
-2x - 1lesqrt{x^2 + 4x + 4 + y^2}\
4x^2+4x + 1 le x^2 + 4x + 4 + y^2\
3x^2-y^2 le 3$



Which seems to be incorrect, what am I doing incorrectly in my procedure? Is my logic correct?







complex-numbers






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asked Dec 1 '18 at 15:43









ChrisKChrisK

1




1












  • $begingroup$
    I can see one wrong step here: From $-2x-1leq sqrt{x^2+4x+4+y^2}$ we cannot deduce the inequality for their squares, that you have written. An example: $-10 leq 3$ but $100notleq 9.$
    $endgroup$
    – user376343
    Dec 5 '18 at 11:37


















  • $begingroup$
    I can see one wrong step here: From $-2x-1leq sqrt{x^2+4x+4+y^2}$ we cannot deduce the inequality for their squares, that you have written. An example: $-10 leq 3$ but $100notleq 9.$
    $endgroup$
    – user376343
    Dec 5 '18 at 11:37
















$begingroup$
I can see one wrong step here: From $-2x-1leq sqrt{x^2+4x+4+y^2}$ we cannot deduce the inequality for their squares, that you have written. An example: $-10 leq 3$ but $100notleq 9.$
$endgroup$
– user376343
Dec 5 '18 at 11:37




$begingroup$
I can see one wrong step here: From $-2x-1leq sqrt{x^2+4x+4+y^2}$ we cannot deduce the inequality for their squares, that you have written. An example: $-10 leq 3$ but $100notleq 9.$
$endgroup$
– user376343
Dec 5 '18 at 11:37










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$begingroup$

Solve the inequality geometrically.



Since $|z-w|$ is the distance between the points representing $z$ and $w$, the inequality $$|z − 2| + |z + 2| ≤ 2$$ can be interpreted as:



The sum of distances from $z$ to $2$ and from $z$ to $-2$ is smaller than $2.$



Put on $x$-axis $-2$ and $2$ and various points $z$ in the plane .... wherever is $zin mathbb{C},$ the sum of distances to $2$ and $-2$ is at least $4,$ obtained when $zin [-2,2].$






share|cite|improve this answer









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    $begingroup$

    Solve the inequality geometrically.



    Since $|z-w|$ is the distance between the points representing $z$ and $w$, the inequality $$|z − 2| + |z + 2| ≤ 2$$ can be interpreted as:



    The sum of distances from $z$ to $2$ and from $z$ to $-2$ is smaller than $2.$



    Put on $x$-axis $-2$ and $2$ and various points $z$ in the plane .... wherever is $zin mathbb{C},$ the sum of distances to $2$ and $-2$ is at least $4,$ obtained when $zin [-2,2].$






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      Solve the inequality geometrically.



      Since $|z-w|$ is the distance between the points representing $z$ and $w$, the inequality $$|z − 2| + |z + 2| ≤ 2$$ can be interpreted as:



      The sum of distances from $z$ to $2$ and from $z$ to $-2$ is smaller than $2.$



      Put on $x$-axis $-2$ and $2$ and various points $z$ in the plane .... wherever is $zin mathbb{C},$ the sum of distances to $2$ and $-2$ is at least $4,$ obtained when $zin [-2,2].$






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        Solve the inequality geometrically.



        Since $|z-w|$ is the distance between the points representing $z$ and $w$, the inequality $$|z − 2| + |z + 2| ≤ 2$$ can be interpreted as:



        The sum of distances from $z$ to $2$ and from $z$ to $-2$ is smaller than $2.$



        Put on $x$-axis $-2$ and $2$ and various points $z$ in the plane .... wherever is $zin mathbb{C},$ the sum of distances to $2$ and $-2$ is at least $4,$ obtained when $zin [-2,2].$






        share|cite|improve this answer









        $endgroup$



        Solve the inequality geometrically.



        Since $|z-w|$ is the distance between the points representing $z$ and $w$, the inequality $$|z − 2| + |z + 2| ≤ 2$$ can be interpreted as:



        The sum of distances from $z$ to $2$ and from $z$ to $-2$ is smaller than $2.$



        Put on $x$-axis $-2$ and $2$ and various points $z$ in the plane .... wherever is $zin mathbb{C},$ the sum of distances to $2$ and $-2$ is at least $4,$ obtained when $zin [-2,2].$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 5 '18 at 11:48









        user376343user376343

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