# How do you solve (x+4)/(1-x)<=0 using a sign chart?

Oct 17, 2016

$x \in \left\{\left(- \infty , - 4\right] , \left(+ 1 , + \infty\right)\right\}$
(see below for method using sign chart)

#### Explanation:

First consider the "boundary values" of $x$ for $\frac{x + 4}{1 - x}$.

The "boundary values" are the values of $x$ for which the numerator or denominator become equal to $0$.

For this example
$\textcolor{w h i t e}{\text{XXX}} x + 4 = 0 \rightarrow x = - 4$
and
$\textcolor{w h i t e}{\text{XXX}} 1 - x = 0 \rightarrow x = 1$

So our boundary values are $\left\{- 4 , + 1\right\}$

and teh sign table looks like
{: (,"||",(-oo,-4),"|",-4,"|",(-4:+1),"|",+1,"|",(+1:+oo),"|"), (,,,,,,,,,,,), (x+4,"||",-ve,"|",0,"|",+ve,"|",+ve,"|",+ve,"|"), (1-x,"||",+ve,"|",+ve,"|",+ve,"|",0,"|",-ve,"|"), (,,,,,,,,,,,), ((x+4)/(1-x),"||",-ve,"|",0,"|",+ve,"|","undef'n","|",-ve,"|"), (,,,,,,,,,,,), ((x+4)/(1-x)<=0,"||","True","|","True","|","False","|","undef'n","|","True","|") :}

So $\frac{x + 4}{1 - x} \le 0 \textcolor{w h i t e}{\text{X}}$ for $x \le - 4 \textcolor{\setminus}{w h i t e} \left(\text{X}\right)$ or $\textcolor{w h i t e}{\text{X}} x > 1$