# Problems choosing the right answer in inequality with variable in denominator [closed]

This question was asked in an exam and I solved it taking into consideration if the denominator is positive or negative. However, there was only one correct answer ($r>0.04$). I have been looking at videos, but I can´t realize why there is only one answer. This is the problem:

Solve for $r$:

$175 > \frac{7}{0.08 - r}$

Thank you in advance

## closed as off-topic by Namaste, Daniel W. Farlow, C. Falcon, Shailesh, LeucippusJun 12 '17 at 0:12

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• It is reasonable to take into consideration the sign of the denominator (since a negative denominator on the right hand side gives a negative ratio, and thus satisfies the inequality). However note that those values $r$ which achieve this are greater than $0.04$ also, so the "one correct answer" incorporates these. Perhaps the only qualification we should add is $r\neq 0.08$ to avoid division by zero. – hardmath Jun 11 '17 at 15:16

If $0.08-r>0$, $$175(0.08-r)>7 \\ 14-175r>7 \\ 175r<7 \\r<\frac{7}{175}\\r<0.04$$
If $0.08 -r<0$, $$175(0.08-r)<7 \\ 14-175r<7 \\ 175r>7 \\r>\frac{7}{175}\\r>0.04$$ But we also have $0.08 -r<0$, therefore $r>0.08$.
We have $$175-\frac{7}{0.08-r}>0$$ or $$\frac{25r-1}{r-0.08}>0,$$ which gives the answer: $r<0.04$ or $r>0.08$.