I have the following inequality linear in $y$

$$ y(Ax+B)(-1)^n-(-1)^n(G+F) < 0 $$

hence $$ y<\dfrac{(-1)^n(G+F)}{(Ax+B)(-1)^n} $$

I would like to know can I omit $(-1)^n$ from denominator and numerator?

I think it is not true to omit it. For example $(-1)^n3x<2(-1)^n$, for $n=2, 3x<2$ and for $n=1$ $-3x<-2 \Rightarrow 3x>2$, which is a different inequality.

Thanks in advance.

  • 2
    $\begingroup$ Your "hence" is not justified. You have divided both sides of an inequality by something that could be negative. $\endgroup$ – Gerry Myerson Feb 11 at 11:43
  • $\begingroup$ @GerryMyerson thank you. So how can I get an inequality on $y$ and the rest of the terms? $\endgroup$ – No one Feb 11 at 11:50
  • 1
    $\begingroup$ You have to deal with various cases, depending on whether $(-1)^n(Ax+B)$ is positive, negative, or zero. $\endgroup$ – Gerry Myerson Feb 11 at 11:55

We have $y(Ax+B)(-1)^n-(-1)^n(G+F) < 0$

If $n$ is even, then we get $y(Ax+B)-(G+F) < 0$,

if $n$ is odd, then we have $G+F - y(Ax+B) < 0$.


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