have a system of 3 nonlinear inequalities in 3 variables, and I need to find x , y , z, in terms of the parameters, that satisfies the equations. The system is: 1 1 + z m − a x &lt; 0 x − b y &lt; 0 y − c z &lt; 0 Where a , b , c &gt; 0 are constants and m a positive integer. How can I go about solving for x, y, z in terms of the parameters?

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have a system of 3 nonlinear inequalities in 3 variables, and I need to find   x  ,  y  ,  z, in terms of the parameters, that satisfies the equations. The system is:                              1                      1            +                          z              m                                      −        a        x        &amp;lt;        0                            x        −        b        y        &amp;lt;        0                            y        −        c        z        &amp;lt;        0            Where   a  ,  b  ,  c  &amp;gt;  0 are constants and m a positive integer. How can I go about solving for x, y, z in terms of the parameters?

Erick Blake · Accepted Answer

Assume the additional constraint   z  &amp;gt;  0.The last two inequalities      {                                    x          −          b          y          &amp;lt;          0                                              y          −          c          z          &amp;lt;          0                        of the given system trap   y by      x    b    &amp;lt;  y  &amp;lt;  c  zso we must have   x  &amp;lt;  b  c  z.Using the first inequality      1          (      1      +              z        m            )        −  a  x  &amp;lt;  0of the given system together with   x  &amp;lt;  b  c  z, we get that   x is trapped by      1          a      (      1      +              z        m            )        &amp;lt;  x  &amp;lt;  b  c  zso a necessary condition on   z is      1          a      (      1      +              z        m            )        &amp;lt;  b  c  zReversing the process, we can solve the given system as follows . . .Let   z be any positive real number such that      1          a      (      1      +              z        m            )        &amp;lt;  b  c  zNote that the above inequality holds for all sufficiently large   z since, as   z approaches infinity, the LHS approaches zero and the RHS approaches infinity.For the chosen   z, choose any   x such that      1          a      (      1      +              z        m            )        &amp;lt;  x  &amp;lt;  b  c  zThen for the chosen   x  ,  z, we have      x    b    &amp;lt;  c  zhence if we now choose any   y such that      x    b    &amp;lt;  y  &amp;lt;  c  zit follows that the system      {                                                1                          (              1              +                              z                m                            )                                −          a          x          &amp;lt;          0                                              x          −          b          y          &amp;lt;          0                                              y          −          c          z          &amp;lt;          0                        is satisfied, and moreover, with the additional constraint   z  &amp;gt;  0, the choices of   x  ,  y  ,  z as described yield all solutions.As an example, the triple   (  x  ,  y  ,  z  ) given by      {                                    x          =                      1            a                                                        y          =                      2                          a              b                                                                    z          =                      3                          a              b              c                                                      always satisfies the given system.

Jude Hunt · Accepted Answer

Got it, that&#039;s extremely helpful, thank you!

have a system of 3 nonlinear inequalities in 3 variables, and I need to find x , y ,

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