The document discusses solving inequalities involving radical functions. It presents two cases: if the index of the radical function is even, then the solutions are all real numbers; if the index is odd, then the solutions are only positive real numbers or zero. The document provides examples of solving various inequalities with radical functions, including determining lower bounds and discussing valid solutions. It also covers combining inequalities with radical functions using rules that depend on whether the index is even or odd.

1. November 28, 2013

2.  to solve inequalities that involve radical
functions

3.  A radical function looks like
n y
radicant
index
 When , we have a square root fuction.
n  2
n  3
 When , we have a cube root function.
 When , we have a fourth root function.
n  4
 When n  5
, we have a fifth root function,
and so on…

4. Given a radical function ,
Case 1:
If n is even, then and
Case 2:
If n is odd, then and
Furthermore,
n y
0 n y  0y 
n y R yR
 n
n y  y

5.  Which of the following equations does not
have a solution? Why?
Because
x  2  0

6.  What can you say about the solutions of the
following inequalities?
4
8
x
3  5  
2
  
3
x
x
2  4  
2

7.  Do the following inequalities have solutions?
What is the lower bound for each of the
inequalities?
4
3 5
3
1 3
x
x
x
 
 
 
x
0   3 
5
4
x
0   
3
x
0   1 
3
Solve the inequalities.

8.  Suppose the following inequalities are valid,
then what must be true about each of the
inequalities?
4 2
6
x x
6 5
3 6
1
x
x
  
  
  
4 2
6
x x
6 0
3 0
0
x
x
 
 
 
Solve the inequalities.

9.  Let’s solve the following inequalities
4
2 3 3
3 2 5
2 4
2
x
x
x
x
 
 
 
 

10.  Up until now, we have discussed the fact
that for
Case 1:
If
k
f x g x
a if g x then
b if g x and f x then f x g x
Case 2:
If
n  2k
2
2
( ) ( )
( ) ( ) 0,
( ) ( ) 0 ( ) 0, ( ) k
( )

 
  
2
k
f ( x ) g ( x
)
a if g x then f x
b if g x then f x g 2
x
( ) ( ) 0, ( ) 0
( ) ( ) 0, ( ) k
( )

 
 

11.  Do you remember the fact that if ,
then and ?
n yR y R 
 Because of this, we have less rules for solving
inequalities whose index is odd.
 In fact,
n  2k 1
k k
2  1 2 
1
f x g x f x g x
( )  ( )  ( ) 
( )
k k
2  1 2 
1
f x g x f x g x
( )  ( )  ( ) 
( )

12.  Solve each of the following inequalities whose
index is odd.
3
 
2 2
3
9 2
1  2  
3
1
1
x
x
x
x

 
Answers:
[1, )
( 14, 14)


 
    
 
 
1
0,
2
x