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2.7
Nonlinear Inequalities
Ex. 1
Solving a Quadratic Inequality
x2 < x + 6
x2 – x – 6 < 0
Now factor.
(x – 3)(x + 2) < 0 3 and –2 are called critical
numbers. Put them on a
3
-2
number line and test each
interval to see if it works.
Pick a number
(
)
bigger than 3.
-2
3
Does it work?
No. Now pick a number between –2 and 3. Does it work?
Yes. Now a number < -2. Does it work?
Ans. (-2,3)
Ex. 2
Solving a polynomial inequality
2x3 + 5x2 > 12x
2x3 + 5x2 – 12x > 0
Take everything to the same
side and factor.
x(2x – 3)(x + 4) > 0
C.N.’s 0, 3/2, -4
(
)
(
-4
0
3/2
3 
 4,0 ,  
2 
2x  7
3
x 5
Ex. 3
2x  7
3 0
x 5
2 x  7  3 x  15
0
x 5
)
5
[
8
Take the 3 over and get
common denominators.
 x 8
0
x5
What are the C.N.’s?
Now put 5 and 8 on
a number line and
test the intervals.
 ,5 8, 
Ex. 4
An inequality involving fractions
x
1
Take everything to one

side.
x2 x3
x
1

 0 Now get common den.
2
x2 x3
x

2
x

2
2
0
x  3x  x  2
 0 ( x  2)( x  3)
( x  2)( x  3) The num. gives imag. roots.
 ,32, 
Therefore, only C.N.’s are
2 and –3.