# when to remove $(-1)^n$ from an inequality

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.

• @GerryMyerson thank you. So how can I get an inequality on $y$ and the rest of the terms? – No one Feb 11 at 11:50
• You have to deal with various cases, depending on whether $(-1)^n(Ax+B)$ is positive, negative, or zero. – 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 $$G+F - y(Ax+B) < 0$$.