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

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)

To solve the inequality (2x - 5) / (3x^4 - x^3 + 275x - 125) >= 0, follow these steps:

Step 1: Find the critical points
To find the critical points, set the numerator and denominator equal to 0.
Numerator: 2x - 5 = 0
Solving this equation: 2x = 5 -> x = 5/2

Denominator: 3x^4 - x^3 + 275x - 125 = 0
Unfortunately, finding the exact solutions for a quartic equation can be quite complex. Thus, we'll use a numerical or graphical method to find the approximate values for the critical points.

Step 2: Use the critical points to create intervals

Now, we have two critical points: x = 5/2 and the two values found for the denominator (let's call them a and b).

Create the intervals on the number line, including the critical points, a, b, and positive/negative infinities. Mark them in ascending order: (−∞, a], (a, b), [b, 5/2), and (5/2, +∞).

This step is crucial because it defines the regions where the inequality is positive or negative.

Step 3: Test the intervals

We need to determine the sign of the expression (2x - 5) / (3x^4 - x^3 + 275x - 125) in each interval to figure out when it is greater than or equal to zero.

Test any value within each interval (except the critical points) and plug it into the inequality expression. If the result is positive or zero, then the interval is part of the solution.

For example:
- Pick a value smaller than 'a', and let's say 'x = -1'. Plugging it into the inequality expression gives us a positive or zero result.
- Pick a value between 'a' and 'b', and let's say 'x = 0'. Plugging it into the inequality expression gives us a negative result.
- Pick a value between 'b' and 5/2', and let's say 'x = 2'. Plugging it into the inequality expression gives us a positive or zero result.
- Pick a value greater than '5/2', and let's say 'x = 3'. Plugging it into the inequality expression gives us a positive or zero result.

Thus, the solution interval in which the inequality (2x - 5) / (3x^4 - x^3 + 275x - 125) >= 0 is satisfied is (-∞, a] ∪ [b, 5/2).

Note: The exact values for 'a' and 'b' depend on the coefficients and factors obtained when solving the denominator equation.