If a . b . c , d are rationals and x is irrational number, then prove that a x + b c x + d is usually a irrational number. When do exceptions occur?

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If   a  .  b  .  c  ,  d are rationals and   x is irrational number, then prove that             a      x      +      b              c      x      +      d       is usually a irrational number. When do exceptions occur?

robegarj · Accepted Answer

The criterion is the following: If   a  d  −  b  c  ≠  0 and   x  ∉            Q       then  y  :=                    a        x        +        b                    c        x        +        d            is irrational as well, but if   a  d  −  b  c  =  0 then y is rational, or undefined.Proof. Assume   c  ≠  0. Then   c  x  +  d is irrational, hence   ≠  0, and we may write  y  =            a      c        −                    a        d        −        b        c                    c        (        c        x        +        d        )                 .If   a  d  −  b  c  ≠  0 then the last fraction, and therewith   y, is irrational. If   a  d  −  b  c  =  0 then   y  =            a      c       is rational.The case   c  =  0 is even simpler: If   d  =  0 as well then   y is undefined. If   d  ≠  0 then   a  d  −  b  c  ≠  0 means   a  ≠  0. Therefore  y  =            a      d        x  +            b      d      is irrational if   a  d  −  b  c  ≠  0, and is   =            b      d      , if   a  d  −  b  c  =  0.

If a . b . c , d are rationals and x is irrational number, then prove

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