Solve the inequality: 2x-10/x >x+5

I tried it and got an unfactorable polynomial: x^2+3x+10, but the answer says it's x<-5 and -2<x<0.

Thanks in advance!

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- Aug 11th 2011, 04:43 PMDragon08Solving Rational Inequalities
Solve the inequality: 2x-10/x >x+5

I tried it and got an unfactorable polynomial: x^2+3x+10, but the answer says it's x<-5 and -2<x<0.

Thanks in advance! - Aug 11th 2011, 04:57 PMe^(i*pi)Re: Solving Rational Inequalities
What is the question and what have you tried?

$\displaystyle \dfrac{2x-10}{x} > x+5$ or is it $\displaystyle 2x - \dfrac{10}{x} > x+5$ - Aug 11th 2011, 06:24 PMSammySRe: Solving Rational Inequalities
- Aug 11th 2011, 06:42 PMDragon08Re: Solving Rational Inequalities
It's the first one, and I made the moved x+5 onto the left side and made a common denominator. The resulting polynomial numerator is unfactorable.

- Aug 11th 2011, 06:43 PMDragon08Re: Solving Rational Inequalities
To SammyS: Oh, then the the book's answer may be wrong.

- Aug 12th 2011, 12:19 AMSironRe: Solving Rational Inequalities
First of all, like SammyS said it has to be -2x in the numerator of the RS, because if it's 2x then you'll get a negative discriminant and so no roots in $\displaystyle \mathbb{R}$.

Rational inequality

$\displaystyle \frac{-2x-10}{x}>x+5$ and $\displaystyle x\neq 0$

Solve:

$\displaystyle \frac{-2x-10}{x}>x+5 \Leftrightarrow -2x-10>x(x+5) \Leftrightarrow -2x-10>x^2+5x \Leftrightarrow -x^2-7x-10>0 \Leftrightarrow -(x+5)(x+2)>0$

Now you can make a sign table to determine the solution. - Aug 12th 2011, 02:57 AMe^(i*pi)Re: Solving Rational Inequalities
Good spot on the -2x instead of 2x (Clap)

Alternatively, if $\displaystyle x < 0$ then we need to change the direction of the inequality when multiplying through by x.

$\displaystyle -2x-10 < x^2+5x \Longleftrightarrow x^2+7x+10 > 0 \Longleftrightarrow (x+2)(x+5) > 0$