Find the minimum value of $|1-1.1y+0.8x|$.

Question

Given $x\le 5$ and $y>5$. I want to minimize th function $|1-1.1y+0.8x|$.

By trial my assumption is $|1-1.1y+0.8x|>0.5$. But I am unable to prove this.

If I'm right then how to prove this ? If I'm wrong then what will be the minimum value ?

Any help plase ?

Answer 1

Given $x\le 5$ and $y>5$. Let $$F=1-1.1y+0.8x$$

$$y>5 \implies -1.1y<-5.5$$ and $$x\le 5\implies 0.8x\le 4$$

So $$F<1-5.5+4 =-0.5\implies |F|>0.5$$

Answer 2

Here is another elementary way:

Set

$$y=5+t \text{ with } t\gt 0 \text{ and } x=5-s \text{ with } s\geq 0$$

That way you get:

\begin{eqnarray*} |1-1.1y+0.8x| & = & |-0.5 - 1.1t-0.8s| \\ & \stackrel{t>0, s\geq 0}{=} & 0.5 + 1.1t + 0.8 s \\ & > & 0.5 \end{eqnarray*}