1. ## Linear Inequality

3/(x-4) < 5 - Wolfram|Alpha

For the inequality on wolfram-alpha, how do you get the algebraic the solution x < 4 ? I only get x > 23/5 when solving algebraically.

2. ## Re: Linear Inequality

Originally Posted by sakonpure6
3/(x-4) < 5 - Wolfram|Alpha

For the inequality on wolfram-alpha, how do you get the algebraic the solution x < 4 ? I only get x > 23/5 when solving algebraically.
Write it as $\displaystyle 0<5-\frac{3}{x-4}=\frac{5x-23}{x-4}$.

Consider the three regions: $\displaystyle \left( { - \infty ,4} \right),\quad \left( {4,4.6} \right),\quad \left( {4.6,\infty } \right)$

$\displaystyle \frac{5x-23}{x-4}$. is positive on the first and third region.

3. ## Re: Linear Inequality

Oh okay, much appreciated!

4. ## Re: Linear Inequality

Originally Posted by sakonpure6
3/(x-4) < 5 - Wolfram|Alpha

For the inequality on wolfram-alpha, how do you get the algebraic the solution x < 4 ? I only get x > 23/5 when solving algebraically.
I believe you have two cases:
1) $\displaystyle x-4 >0$
2) $\displaystyle x-4 <0$

You've only worked out case 1.

For case 2, you end up with $\displaystyle x<\frac{23}{5}$. This is a condition and is satisfied whenever $\displaystyle x-4<0$.
Hence, in the case where $\displaystyle x-4<0$, the inequality is satisfied by all such x. That's where the $\displaystyle x<4$ comes from.