section 1.6
DESCRIPTION
Section 1.6. Polynomial and Rational Inequalities. Polynomial Inequalities. We said that we can find the solutions (a.k.a. zeros) of a polynomial by setting the polynomial equal to zero and solving. We are going to use this skill to solve inequalities such as:. Solving Quadratic Inequalities. - PowerPoint PPT PresentationTRANSCRIPT
Section 1.6
Polynomial and Rational Inequalities
Polynomial Inequalities
We said that we can find the solutions (a.k.a. zeros) of a polynomial by setting the polynomial equal to zero and solving.
We are going to use this skill to solve inequalities such as:
0122 xx
Solving Quadratic Inequalities
0122 xx
034 xx
034 xx
Factor
Identify the zeros (critical points)
There are now 3 intervals: (-∞,-3), (-3,4), and (4,∞).
We will test these three intervals to see which parts of this function are less than (negative) or greater than (positive) zero.
4x 3x
Testing Intervals
To test, pick a number from each interval and evaluate
Instead of evaluating, we can also just check the signs of each factor in our factored form of the polynomial.
034 xx
Solution: (-∞,-3) U (4,∞)
Recap of Steps
Factor and solve the quadratic to find the critical points
Test each intervalDetermine if (+) or (-) values are desired
253 2 mm
0253 2 mm
0213 mm
23
1 andm
Solve the Inequality
Solution:
x2 – 2x ≥ 1
Solution:
0122 xx
12
11422 2 x
2
82 x
2
222 x
21x
4.04.2 andx
,2121,
x2 + 2x ≤ -3
0322 xx
12
31422 2 x
2
82 x
21 ix
No Real Solutions
Test any number to find out if all numbers are true or false.
Solving Rational Inequalities
064
12
x
x
088
1
xx
x
Solution: (-∞,-8) U (-1,8)
1x 8x 8x
8x
-8 -1 8
Restrictions?