Let's Learn to Solve Inequalities!

\(\frac{x^2+7x+2}{x-2}>2\)

Take 2 to left hand side, we get

\(\frac{x^2+7x+2}{x-2} - 2 > 0\)
=> \(\frac{x^2+7x+2 - 2*(x-2) }{x-2}>0\)
=> \(\frac{x^2+7x+2 - 2x + 4 }{x-2}>0\)
=> \(\frac{x^2+5x+6}{x-2}>0\)
=> \(\frac{x^2+2x + 3x +6}{x-2}>0\)
=> \(\frac{x(x+2) + 3(x +2)}{x-2}>0\)
=> \(\frac{(x+2)* (x+3)}{x-2}>0\)
Multiply numerator and denominator on left hand side by x-2 we get
=> \(\frac{(x+2)* (x+3) * (x-2)}{(x-2)^2}>0\)
Denominator is \((x-2)^2\) which will anyways be > 0 so we can ignore this to solve the problem
So, question reduces to (x+2)* (x+3) * (x-2) > 0

Follow this post to learn about Sine curve method of solving the problem to follow the method

Points on the sign curve will be -3, -2, 2 [ we get this by putting x+3 = 0, x+2=0, x-2 = 0]
See below image




Since question is asking for > 0 so answer will be the values indicated in the "+" region in the above image
So, answer is -3<x<-2 and x>2.

Check answer by substituting and tell me if it works!
Hope it helps!


Watch the following video to learn How to Solve Inequality Problems