Hello all,
Can someone please help me with the following proof? i have an exam on monday and cant figure out this one question i found online:
Use order axioms to show that 0 < a < b & 0 < c < d implies that
a/d < b/c. Thanks in advance.
Hello all,
Can someone please help me with the following proof? i have an exam on monday and cant figure out this one question i found online:
Use order axioms to show that 0 < a < b & 0 < c < d implies that
a/d < b/c. Thanks in advance.
I hope I remember: $\displaystyle \frac{a}{d}<\frac{b}{c} \Longleftrightarrow ac<bd$ $\displaystyle \Longleftrightarrow ac-bd <0 \Longleftrightarrow ac-ad+ad-bd<0$ $\displaystyle \Longleftrightarrow a(c-d)+d(a-b)<0$, and since the last inequality is exactly the opposite that we have by the given data ($\displaystyle c<d \Longrightarrow c-d<0\,,\,\,a<b \Longrightarrow a-b<0$ and everybody's positive here, the correct inequality should be $\displaystyle \frac {a}{d}>\frac{b}{c}$
Tonio