Conclusion The solution set of the given rational inequality is given by the interval
( ∞ , 2) U (1 , + ∞)
Graphical solution to the given inequality.
Below is shown the graph fo the function y = (x1)/(x+2). It is easy to check graphically that y = (x1)/(x+2) is positive over the interval
( ∞ , 2) U (1 , + ∞).
Example 3 Solve the rational inequality given by
Solution to Example 3
NOTE: Do not multiply both sides by the LCD (x  3)(x + 4)
as you would do if this was an equation. The sign of (x  3)(x + 4) changes with x
and we do not know if the order of the inequality is to be changed or not.
Given
Write the inequality with the
right side equal to zero by subtracting
3 / (x + 4) from both sides.
Rewrite the inequality so
that the two terms making the left side have common denominator.
Multiply factors, add the two rational expressions
on the left side of the inequality and group like terms in the numerator to obtain
The zero of the numerator:  x + 17 = 0 is x = 17
The zeros of the denominator: (x  3)(x + 4) = 0 are x = 3 and x = 4
The three zeros divide the real number line into four intervals
(Note that the zeros are ordered from the smallest to the largest).
(∞ , 4) , (4 , 3) , ( 3 , 17) , (17 , +∞)
a) ( ∞ , 4) test value: x = 5
We now evaluate ( x + 17) / [ (x  3)(x + 4) ] at x = 5 to find its sign.
( x + 17) / [ (x  3)(x + 4) ]
= ( (5) + 17) / [ (5  3)(5 + 4) ]
= 22/8 (positive)
b) ( 4 , 3) test value: x = 0
We now evaluate ( x + 17) / [ (x  3)(x + 4) ] at x = 0 to find its sign.
( x + 17) / [ (x  3)(x + 4) ]
= ( 0 + 17) / [ (0  3)(0 + 4) ]
= 17 / 12 (negative)
c) (3 , 17) test value: x = 4
We now evaluate ( x + 17) / [ (x  3)(x + 4) ] at x = 4 to find its sign.
( x + 17) / [ (x  3)(x + 4) ]
= ( 4 + 17) / [ (4  3)(4 + 4) ]
= 13/8 (positive)
d) (17 , +∞) test value: x = 18
We now evaluate ( x + 17) / [ (x  3)(x + 4) ] at x = 18 to find its sign.
( x + 17) / [ (18  3)(x + 4) ]
= ( 18 + 17) / [ (18  3)(18 + 4) ]
= 1/330 (negative)
Let us now put all the above results in a table
∞ 
4 

3 

17 
∞ 
+ 
undefined 
 
undefined 
+ 
0 
 
Conclusion The solution set of the given rational inequality
is given by the interval
( 4 , 3) U (17 , + ∞)
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