4.6.1 Polynomial and Rational Inequalities - Part I
Example:
x3 + 2x2 - 8 < 0
Objective: describe all values of x that satisfy the inequality
Principle: a polynomial f(x), when graphed, is a continuous
curve:




it crosses the x-axis at its zeros, i.e. where f(x) = 0)
these zeros are called boundary numbers
and are the only points where f(x) can change signs
the boundary numbers divide up the whole real axis into
several (in this case, 4) intervals
 throughout each interval, f(x) has the same sign, i.e.
always < 0 or > 0
Interval
(-, zero 1)
(zero 1, zero 2)
(zero 2, zero 3)
(zero 3, )
Key idea:
Sign of f(x)
f(x) > 0
f(x) < 0
f(x) > 0
f(x) < 0
locate the zeros (boundary numbers)
4.6.1-1
Method
2
Example: x + 2x < 8
1. get everything on lefthand side, 0 on right
2. determine zeros of
poly.
3. locate the zeros on a
number line. Use
open bubbles (since
the inequality is
strict). The zero will
not be part of the
solution.
4. tabulate the intervals.
x2 + 2x - 8 < 0
(x - 2)(x + 4) = 0
x = 2, -4
----------------------------------4
2
Interval
Test Value x
f(x)
(-, -4)
(-4, 2)
(2, )
5. choose a test value on Interval Test Value x f(x)
7
each interval, and
(-, -4) -5
evaluate polynomial
(-4, 2) 0
-8
for each test value
3
7
(2, )
(use GC TABLE
facility)
6. observe direction of
f(x) < 0 for the middle interval
inequality from 1 and ans: (-4, 2)
the bubble style.
State the answer.
Note: if the original inequality were "" instead of "<",
bubbles would be solid, and answer would be [-4, 2].
4.6.1-2