4.6.2 Polynomial and Rational Inequalities - II
rational expression inequality:
x 2  3x  10  0
1 x
Principle:
 much like polynomial inequalities
 can change signs at zeros of numerator
 but rational expression can change signs at zeros of
denominator (where its vertical asymptotes are)
 boundary numbers: all zeros of numerator and denominator
Example and method:
x 2  3x  10  2
1 x
Preparation: use algebra to write inequality in standard
form:
p( x)
on left-hand side of inequality, 0 on right-hand side
q( x)
BUT NEVER multiply or divide both sides by an
expression involving the unknown! Do you know why?
Here’s what you need to do:
x 2  3x  10 - 2  0
(everything on LHS)
1 x
x 2  3x  10  2(1  x )
 0 (review pp. R-41-R-43)
1 x
x 2  x  12  0
(standard form)
1 x
4.6.2-1
Method
2
x
 3x  10  2
Example:
1 x
1. get everything on leftx 2  x  12  0 (prev. page)
hand side, 0 on right
1 x
2. determine zeros of
(x + 3)(x - 4) = 0 1 - x = 0
numerator and
x = -3, 4
x=1
denominator
3. locate the zeros on a
-------------------------------------number line: solid
-3
1
4
bubbles for those
zeros that will be part
of the solution
Interval Test Value x f(x)
4. tabulate the intervals
(-, -3)
(-3, 1)
(1, 4)
(4, )
5. choose a test value on Interval Test Value x f(x)
1.6
each interval, and
(-, -3) -4
evaluate the rational
(-3, 1) 0
-12
expression (see 1) for (1, 4)
2
10
each test value (use
5
-2
(4, )
GC TABLE facility)
7. observe direction of
f(x)  0 for the 1st and 3rd intervals
inequality from 1.
ans: (-, -3]  (1, 4]
State the answer,
observing bubbles.
4.6.2-2