Express the following equation, using O ? or T notation

Question

Express the following equation, using Ω Ο or Θ notation
2|x| ≤ | ((4x-5)(2x+5))/(3x^2+6) | ≤ 18|x|

Answer 1

You have upper and lower bounds that are multiples of |x|. That means the function in the middle is in theta(|x|). Usually in computer science we only care about the behavior as x goes to infinity, where |x|=x, so a simpler expression would be that the rational function is in theta(x).