See where it's equal and where it's undefined.

Undefined if x = 0

Cross multiply: x(x - 3) = -x + 8

x^2 - 3x = -x + 8

x^2 - 2x - 8 = 0

(x - 4)(x + 2) = 0

x = 4 or x = -2

This sets up four intervals: x ≤ -2; -2 ≤ x < 0; 0 < x ≤ 4, and x ≥ 4

Test a number in each interval to see if it works in the original. The intervals where it comes out true are your answer. For example, to test x ≤ -2 plug in -3:

(-3 + 8) ÷ -3 ≥ -3 - 3

-5/2 ≥ -6 true

To test -2 ≤ x < 0 try x = -1:

(-1 + 8) ÷ -1 ≥ -1 - 3

-7 ≥ -4 false

(-x+8)/x ≥ (x - 3)

Subtract x - 3 from both sides:

(-x+8)/x - x + 3 ≥ 0

Get a common denominator of x:

(-x+8)/x - x²/x + 3x/x ≥ 0

Combine the numerators:

(-x² - 2x + 8)/x ≥ 0

Factor:

-(x² + 2x - 8)/x ≥ 0

-(x + 4)(x - 2)/x ≥ 0

The important points are are x=-4, x=0, x=2.

The places where the result is ≥ 0 are:

x ≤ -4 or 0 < x ≤ 2

Answer:

x ≤ -4 or 0 < x ≤ 2