Review:
A polynomial can change sign only at its real zeros. When ordered, the real zeros of a
polynomial divide the real number line into
intervals in which the polynomial does not change sign.
Example 1: Solve the polynomial inequality
x2 < -x + 6
Solution to Example 1:
- Given
x2 < -x + 6
- Rewrite the inequality with one side equal
to zero.
x2 + x - 6 < 0
- Factor the left side of the inequality.
(x - 2)(x + 3) < 0
- The two real zeros -3 and 2 of the left side of the inequality, divide the
real number line into 3 intervals.
(-? , -3) (-3 , 2) and (2 , +?)
- The sign within each interval is
determined by using test values. We chose one value within each interval and use
it to find the sign of (x - 2)(x + 3).
- a) (-? , -3)
- chose x = -4 and evaluate (x - 2)(x + 3)
(x - 2)(x + 3) = (-4 - 2)(-4 + 3)
= 6
(x - 2)(x + 3) is positive in (-? ,
-3)
- b) (-3 , 2)
- chose x = 0 and evaluate (x - 2)(x + 3)
(x - 2)(x + 3) = (0 - 2)(0 + 3)
= -6
(x - 2)(x + 3) is negative in (-3 , 2)
- c) (2 , +?)
- chose x = 4 and evaluate (x - 2)(x + 3)
(x - 2)(x + 3) = (4 - 2)(4 + 3)
= 14
(x - 2)(x + 3) is positive in (2 , +?)
- We now put all the above information in a
table.
Conclusion
We are looking for values of x that make (x - 2)(x + 3)
negative. The solution set consists of all real numbers in the
interval (-3 , 2).
Matched Exercise:Solve the polynomial inequality
x2 + 4x < 5
Example 2: Solve the polynomial inequality
(x2 + 2)(x + 1)(x + 6) > 0
Solution to Example 2:
Conclusion
The solution set consists of all real numbers in (- ? , -6) U (-1 , + ?).
Matched Exercise: Solve the polynomial inequality
-(x6 + 8)(x - 1)(x + 4) < 0
More references and links on how to Solve Equations, Systems of Equations and Inequalities.
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