009. Graphing Inequalities

For real numbers aa, bb, and cc such that 0.9a1.10.9 \leq a \leq 1.1, 2.7b3.32.7 \leq b \leq 3.3, and 4.5c5.44.5 \leq c \leq 5.4, show the range of a point P(u,v)P(u, v) moving where u=b/au = b/a and v=c/av = c/a.

It seems easy at first glance. 10111a109    2711ba113,4511ca6\begin{aligned} \frac{10}{11} \leq \frac{1}{a} \leq \frac{10}{9} \implies \frac{27}{11} \leq \frac{b}{a} \leq \frac{11}{3}, \quad \frac{45}{11} \leq \frac{c}{a} \leq 6 \end{aligned}

However, be warned that the real range is not a rectangle region as the above result. Another condition must be considered as well. 10331b1027,vu=cb    1511vu2\begin{aligned} \frac{10}{33} \leq \frac{1}{b} \leq \frac{10}{27}, \quad \frac{v}{u} = \frac{c}{b} \implies \frac{15}{11} \leq \frac{v}{u} \leq 2 \end{aligned}

It provides the 2 additional equations of lines. Since uu is positive, 1511uv2u\begin{aligned} \frac{15}{11}u \leq v \leq 2u \end{aligned}


Another point of view of this problem is scaling. uu and vv are scaled by 1/a1/a and their original unscaled range can be considered as 2.7u3.32.7 \leq u \leq 3.3 and 4.5v5.44.5 \leq v \leq 5.4. In other words, the original rectangle region of uu and vv is scaled by the range of 1/a1/a and the range of PP is where this region slides.


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