“In third grade, I cheated on my history exam. In fourth grade,
I stole my uncle Max's toupee and I glued it on my face when
I was Moses in my Hebrew School play. In fifth grade, I
knocked my sister Edie down the stairs and I blamed it on the
dog....”
3.5 Polynomial and Rational Inequalities
Solving Equations
Let’s review solving the following equation:
2x  x  6  0
2
f (x)  2x 2  x  6

(-1.5,0)
(2,0)
2x 2  x  6  0
*A polynomial function can only change signs at its zeros (x-ints).*

Solving Inequalities
Solve the following inequality without graphing:
6x  6  5x
2
Solving Polynomial Inequalities
What steps did we use to solve 6x 2  6  5x ?
6x 2  5x  6  0
1) Get everything on the left side to set equal to zero.
(2x  3)(3x  2) 0 Zeros : (2 3,0),(3 2,0)
2) Find the zeros of f (x) by factoring.

32
2 3
f (1)  
f (0)  
f (2)  
3) Use zeros to separate real number line into test intervals.


4) Pick value
from each
testinterval to evaluate sign of f (x).


(,2 3][3 2,)
5) Write out solution region in interval notation.
Solving Inequalities
Solve the following inequality without graphing:
x  4 x  4 x 16
3
2
Solving Inequalities
Solve the following inequality without graphing:
x 3  3x 2
Solving Inequalities
Solve the following inequalities without graphing:
1) x 3  2x 2  3x  0
2) x 4
 13x  36  0
2
Solving Rational Equations
Let’s review solving the following equation:
4x  5
0
x2
4x  5
f (x) 
x2
4x  5
0
x2


(-1.25,0)
*A rational function can change signs at its
zeros (x-ints) AND where f is undefined.*
Solving Inequalities
Solve the following inequality without graphing:
3  3x
2
2x  5
Solving Rational Inequalities
What steps did we use to solve rational inequalities?
3  3x
2 0
2x  5
1) Get everything on the left side and zero on the other.
7x  7
 0 Zero : (-1,0) DNE : x  5 2
2x  5
2) Find the
zeros AND where f (x) DNE by finding common
denominator.
1
5 2
f (3)  
f (0)  
f (1)  
3) Use these to separate real
number line into test intervals.


4) Pick value
test interval
to evaluate sign of f (x).
 from each


(5 2,1]
5) Write out solution region in interval notation.
Solving Inequalities
Solve the following inequality without graphing:
5
3

x  3 x 1
Solving Inequalities
Solve the following inequality without graphing:
15
x 8
x
Practice Problem
Solve the following inequality without graphing:
x4
1
2x  4
3.5 Polynomial and Rational Inequalities
Homework #3:
Page 217
#11, 17, 23, 33 – 37 Odd, 43, 45, 49