solve the rational inequality

Question

solve the rational inequality 2-5/x+2/x^2>0

Answer 1

x²(2-5/x+2/x²)>0(x²⁾

2x²-5x+2>0

(2x-1)(x-2)>0  Equation A

2x-1 = 0 or x-2 = 0

   2x = 1 or     x = 2

     x = 1/2  or x = 2

Set up a table in the following way:

 

(-∞,1/2)    (1/2, 2)    (2,∞)

 

Now choose a number between -∞ and 1/2

Let's choose x = 0 and substitute that into Equation A

 

(2(0)-1)(0-2)= (-1)(-2)=2  You only care that it is a positive number

Place a plus sign under (-∞,1/2) on your table.

 

Next, choose a number between 1/2 and 2.

Let's choose x = 1 and substitute that into Equation A

(2(1)-1)(1-2)=(1)(-1)=-1  Again you only care that it is a negative number

 

Last, choose a number between 2 and ∞

Let's choose x = 4 and substitute into Equation A

(2(4)-1)(4-2)=(7)(2)=14.  So, your answer is positive.

 

Now your table looks like this:

 

(-∞,1/2)    (1/2, 2)    (2,∞)

 

     +               -              +

 

Since your original has a > sign, you only want those positive results.

Answer: (-∞,1/2) U (2,∞)