Question

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

Answer 1

`x <-7 vvx>3`

Explanation:

This is a rational inequation where `(P(x))/(Q(x))>0`
Therefore we have to solve:

`P(x)>0`
`Q(x)>0`

And plot the sign chart

`x+7>0=>x > -7`
`x-3>0=>x>3`

Therefroe the follows regions of xy-plot

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

Answer 2

`(x+7)/(x-3) > 0` for `x < -7` and `x > 3`

Explanation:

Given `(x+7)/(x-3)`

The obvious points of interest will occur when
`color(white)("XXX")x+7= 0 rarr x=-7`
and when
`color(white)("XXX")x-3=0rarr x=3`

This gives us 3 ranges;
for each we can ask if the relation `(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") :}#