# How do you solve (x+7)/(x-3)>0 using a sign chart?

Oct 5, 2016

$x < - 7 \vee x > 3$

#### Explanation:

This is a rational inequation where $\frac{P \left(x\right)}{Q \left(x\right)} > 0$
Therefore we have to solve:

$P \left(x\right) > 0$
$Q \left(x\right) > 0$

And plot the sign chart

$x + 7 > 0 \implies x > - 7$
$x - 3 > 0 \implies x > 3$ Therefroe the follows regions of xy-plot

graph{(x+7)(x-3)>0 [-8.89, 8.885, -4.444, 4.44]}

Oct 5, 2016

$\frac{x + 7}{x - 3} > 0$ for $x < - 7$ and $x > 3$

#### Explanation:

Given $\frac{x + 7}{x - 3}$

The obvious points of interest will occur when
$\textcolor{w h i t e}{\text{XXX}} x + 7 = 0 \rightarrow x = - 7$
and when
$\textcolor{w h i t e}{\text{XXX}} x - 3 = 0 \rightarrow x = 3$

This gives us 3 ranges;
for each we can ask if the relation $\frac{x + 7}{x - 3} > 0$ is true.

Sign chart
{: ("range: ",x < -7,color(white)("X"),x in (-7,+3),color(white)("X"),x > +3), ("sample value in range: ",-10,,0,,+10), ((x+7)/(x-3)" at sample value: ",+ve,,-ve,,+ve), ((x+7)/(x-3) > 0,"Yes",,"No",,"Yes") :}