Question

Solve the following inequality and write answer using interval notation. Please show all of your work.
2x-5/3x^4-x^3+275x-125>=0

Answer 1

Your casual use of parentheses makes me assume you mean

(2x-5)/(3x^4 - x^3 + 275x - 125) >= 0
let's try to factor the denominator. We get not much help.

You sure you don't mean

3x^4 - x^3 + 375x - 125?

That would be

x^3 (3x-1) + 125(3x-1)
(x^3+125)(3x-1)
(x+5)(x^2 - 25x + 25)(3x-1)

Anyway, working with your original function, the denominator is positive when x < -4.54 or x > 0.45

The numerator is positive when x > 2.5

If you plot the points -4.54, 0.45, 2.5 on the number line, you will see

on (-oo,4.54) we have +/- so y < 0
on (-4.54,0.45) we have -/- so y > 0
on (0.45,2.5) we have +/- so y < 0
on (2.5,oo) we have +/+ so y > 0

y>0 on [-4.54,0.45] U [2.5,oo)