# Math Help - solving rational equation

1. ## solving rational equation

hello, I cannot seem to understand the process for solving this.
Can somone please show me the steps involved.

Thanks for any help.

(1/x)+5 > 1/3x

I was thinking:
1/x > (1/3x)-5
(1/x) - 1/3x > -5
1/x > -5
x > 5

but I know my attempt is incorrect.

2. ## Re: solving rational equation

Originally Posted by fran1942
hello, I cannot seem to understand the process for solving this.
Can somone please show me the steps involved.

Thanks for any help.

(1/x)+5 > 1/3x

I was thinking:
1/x > (1/3x)-5
(1/x) - 1/3x > -5
1/x > -5
x > 5

but I know my attempt is incorrect.
Is the inequality $\displaystyle \frac{1}{x} + 5 > \frac{1}{3}x$ or $\displaystyle \frac{1}{x} + 5 > \frac{1}{3x}$?

3. ## Re: solving rational equation

It is the second equation you quoted.

thanks.

4. ## Re: solving rational equation

First note that $\displaystyle x \neq 0$.

\displaystyle \begin{align*} \frac{1}{x} + 5 &> \frac{1}{3x} \\ \frac{1}{x} - \frac{1}{3x} &> -5 \\ \frac{3}{3x} - \frac{1}{3x} &> -5 \\ \frac{2}{3x} &> -5 \end{align*}

Now you need to consider two cases, $\displaystyle x < 0$ and $\displaystyle x > 0$.

Case 1:

\displaystyle \begin{align*} \frac{2}{3x} &> -5 \\ 2 &< -15x \\ -15x &> 2 \\ x &< -\frac{2}{15} \end{align*}

And since we know $\displaystyle x < 0$, that means $\displaystyle x < -\frac{2}{15}$ is acceptable.

Case 2:

\displaystyle \begin{align*} \frac{2}{3x} &> -5 \\ 2 &> -15x \\ -15x &< 2 \\ x &> -\frac{2}{15} \end{align*}

And since we know $\displaystyle x > 0$, that means $\displaystyle x > 0$ is acceptable.

So the solution is $\displaystyle x \in \left(-\infty, -\frac{2}{15}\right) \cup (0, \infty)$