SOLUTION: Textbook Beecher, Penna it is an exercise on polynomials and rational inequalities I think my answer is -4 but I'm not sure. thank you for your help Solve 4>=4/x+x

x cannot be 0 since we have the term  and
we cannot divide by 0. So there are two cases:

Case 1:  (i.e., x is positive)

Since x is in this case a positive number, if we
clear of fractions by multiplying through by positive
number x, the inequality symbol will NOT be reversed:

 AND 

 AND 

 AND 

 AND 

Multiply through by -1 to make the first term positive.
This DOES reverse the inequality symbol:

 AND 

 AND 

 AND 

Since the square of a real number is never negative,
the left side can never be less than 0, so it
can only equal 0:

 AND 
 AND 
 AND 

So we get only one solution when ,

namely x = 2
 
Case 2:  (i.e., x is negative)

Since x is in this case a negative number, if we
clear of fractions by multiplying through by positive
number x, the inequality symbol WILL be reversed:

 AND 

 AND 

 AND 

 AND 

Multiply through by -1 to make the first term positive.
This DOES reverse the inequality symbol:

 AND 

 AND 

 AND 

Since the square of any real number is never negative,
the first part  will be true for all
values of x, and the second part  is the
only thing that must be satisfied.  So the solution is

 

The graph of this is


<=============o-----@------>
 -4 -3 -2 -1  0  1  2  3  4 

where "o" means "an open circle" and where "@" means a
closed circle.  This is abbreviated in interval 
notation as

(,0) U {2}

Edwin