Solve the following rational inequality. Write the solution in interval notation. Show the number line and breaking points or graph of the polynomial with the part of the graph that is the solution (x-4)(x+2)/(x-3)(x+5) ¡Ü0

Answer 1

I have a strong feeling you meant
(x-4)(x+2)/( (x-3)(x+5) ) ≥ 0

I good and fast way is to look at the graph of
y = (x-4)( (x+2)/(x-3)(x+5) )

http://www.wolframalpha.com/input/?i=y+%3D+%28x-4%29%28x%2B2%29%2F%28%28x-3%29%28x%2B5%29%29

It shows the graph to be above the x-axis , ( > 0 ) , for x < -5, between -2 and 3 and then again for x > 4
However we must exclude x = 3 and x = -5, since we have vertical asymtotes at these two values

so if your symbol is to be ≥ , then

-2 ≤ x < 3 OR x ≥ 4

another way to algebraically analyse it .
The "critical " values of x are -5, -2, 3, and 4
splitting up the x-axis into 5 sections.
I will then arbitrarily pick any value in each region and calculate using only the ± signs of the factors. We don't really care what the actual answer is, we only care about whether it is + (above) or - (below)

1. for any x < -5
let x = -6, I get (-)(-)/((-)(-) > 0 , that's good
2. for -5<x<-2
let x = -4, I get (-)(-2)/((-)(+)) < 0 , no good
3. for -2 < x < 3
let x = 0 , I get (-)(+)/((-)(+)) > 0 , that's good
4. for 3 < x < 4
let x = 3.5, I get (-)(+)/((+)(+) < 0 , no good
5. for x > 4
let x = 10 , I get (+)(+)/((+)(+)) > 0 , that's good again

so we have:
x < -5 OR -2≤ x < 3 OR x ≥ 4

Answer 2

less than or equal to