1. 2-4: Other Types of
Equations
© 2007 Roy L. Gover (www.mrgover.com)
Learning Goals:
•Solve absolute value
equations
•Solve radical equations
•Solve fractional equations

2. Important Idea
In the last lesson, we
solved linear and quadratic
equations. In this lesson,
we are going to solve other
kinds of equations
important in mathematics.

3. Definition
The absolute value of a
number c is denoted and
is defined as follows:
c
If , then0c ≥ c c=
If , then0c < c c= −

4. Example
If , then0c ≥ c c=
If , then0c < c c= −
Using the definition of
absolute value above, find
the absolute value when:
2c = 3c = − 0c =

5. Definition
If c and d are real numbers,
then is the distance
between c and d on the
number line.
c d−

6. Example
0 10-10
Find the distance between
10 and -10:
10 ( 10) 10 10 20 20d = − − = + = =
20

7. Example
0 10-10
Find the distance between
-10 and 10:
10 (10) 10 10 20 20d = − − = − − = − =
20

8. Try This
True of False?
c d c d+ = +
Give an example to
support your answer.
False; 2c = & 1d = −

9. 4 8x − =
Important Idea
The following statements
mean the same thing:
The distance from x to 4 is 8
units.

10. Example
Solve 4 8x − =

11. Example
Solve 2 4 8x − =
Confirm your answer(s)
with your calculator.

12. Try This
4 2 8x − =Solve
5 3
,
2 2
x = −
Confirm with your calculator.

13. Try This
4 2 8x − = −
Solve
Confirm with your calculator.
?

14. Example
2
4 3 2x x+ − =
Solve using the algebraic
definition of absolute value:

15. Definition
Radical equations are
equations that contain
expressions under a radical
symbol.

16. Example
5 3 11x x+ − =
Solve:

17. Important Idea
It is always a good idea to
verify solutions (algebraically
or graphically). Solutions to
radical equations must be
verified. Solutions which turn
out not to be solutions are
called extraneous roots.

18. Try This
Solve. Check for
extraneous roots:
3 2 7x − =
17x =

19. Definition
The Power Principle is the
process of raising both sides
of an equation to the same
power. It is what we have
been doing to solve radical
equations. Sometimes the
Power Principle must be
applied more than once.

20. Example
Solve algebraically. Be sure
to check for extraneous
roots.
2 3 7 2x x− − + =

21. Try This
Solve algebraically. Be sure
to check for extraneous
roots.
2 2 3 1x x+ − − =
x=2

22. Definition
Fractional Equations,
sometimes called Rational
Equations, are equations in
the form:
( )
0
( )
f x
g x
=
where ( ) 0g x ≠

23. Important Idea
To solve fractional equations
of the form:
( )
0
( )
f x
g x
=
find all solutions where
( ) 0g x ≠( ) 0f x = and

24. Example
Solve. Check for
extraneous roots.
2
2
6 1
0
2 9 5
x x
x x
− −
=
+ −

25. Try This
Solve. Check for
extraneous roots.
2
2
2 3
0
2 5 3
x x
x x
+ −
=
+ +
1x =

26. Lesson Close
The ability to solve all
types of equations is one of
the most basic skills in
mathematics.