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?