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Question Number 1335 by 123456 last updated on 23/Jul/15
if a≤x≤b and c≤y≤d then did  (a/d)≤(x/y)≤(b/c)?  a,b,c,d,x,y∈R  0<c  0<a
$$\mathrm{if}\:{a}\leqslant{x}\leqslant{b}\:\mathrm{and}\:{c}\leqslant{y}\leqslant{d}\:\mathrm{then}\:\mathrm{did} \\ $$$$\frac{{a}}{{d}}\leqslant\frac{{x}}{{y}}\leqslant\frac{{b}}{{c}}? \\ $$$${a},{b},{c},{d},{x},{y}\in\mathbb{R} \\ $$$$\mathrm{0}<{c} \\ $$$$\mathrm{0}<{a} \\ $$
Answered by prakash jain last updated on 23/Jul/15
a≤x  ay≤xy     ∵y>0  ay≤xd     ∵d≥y  (a/d)≤(x/y)      ∵dy>0  x≤b  xc≤bc       ∵c>0  xc≤by        ∵y≥c  (x/y)≤(b/c)        ∵yc>0
$${a}\leqslant{x} \\ $$$${ay}\leqslant{xy}\:\:\:\:\:\because{y}>\mathrm{0} \\ $$$${ay}\leqslant{xd}\:\:\:\:\:\because{d}\geqslant{y} \\ $$$$\frac{{a}}{{d}}\leqslant\frac{{x}}{{y}}\:\:\:\:\:\:\because{dy}>\mathrm{0} \\ $$$${x}\leqslant{b} \\ $$$${xc}\leqslant{bc}\:\:\:\:\:\:\:\because{c}>\mathrm{0} \\ $$$${xc}\leqslant{by}\:\:\:\:\:\:\:\:\because{y}\geqslant{c} \\ $$$$\frac{{x}}{{y}}\leqslant\frac{{b}}{{c}}\:\:\:\:\:\:\:\:\because{yc}>\mathrm{0} \\ $$

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