Please Help with this question Solve the rational inequality

Please Help with this question...

Solve the rational inequality indicated using a number line and the behavior of the graph at each zero. 1/x + 3

Solution

1 / ( x+ 3) < ( x- 5 ) / (x - 11)

or

1/ ( x+ 3) - (x-5) / (x-11) <0

( x- 11) - (x-5)(x+3) / (x+3)(x-11) < 0

( x- 11 - x^2 + 2x + 15 ) / ( x+ 3)(x-11) < 0

(-x^2 + 3x + 4 ) / ( x+3)(x-11) < 0

critical points are -3 , -1 , 4 , 11

testing the inequality at 5 intervals

---------------- -3 ---------- -1 ----------------4------------11----------

( -infinity to -3) test point -4

(-x^2 + 3x + 4 ) / ( x+3)(x-11) < 0

-4^2 + 3(-4) + 4 / ( -4+3)(-4-11) < 0

true

2nd interval (-3 to -1 ) test point -2

false

3rd interval ( -1 to 4)   test point 0

true

4th interval ( 4 to 11) test point 5

false

5th interval ( 11 to infinity ) test point 12

true

solution in interval notation is

( -infinity , -3 ) U ( -1 to 4 ) U ( 11 to infinity )