2-4: Other Types of
Equations
© 2007 Roy L. Gover (www.mrgover.com)
Learning Goals:
•Solve absolute value
equations
•Solve...
Important Idea
In the last lesson, we
solved linear and quadratic
equations. In this lesson,
we are going to solve other
k...
Definition
The absolute value of a
number c is denoted and
is defined as follows:
c
If , then0c ≥ c c=
If , then0c < c c= −
Example
If , then0c ≥ c c=
If , then0c < c c= −
Using the definition of
absolute value above, find
the absolute value when...
Definition
If c and d are real numbers,
then is the distance
between c and d on the
number line.
c d−
Example
0 10-10
Find the distance between
10 and -10:
10 ( 10) 10 10 20 20d = − − = + = =
20
Example
0 10-10
Find the distance between
-10 and 10:
10 (10) 10 10 20 20d = − − = − − = − =
20
Try This
True of False?
c d c d+ = +
Give an example to
support your answer.
False; 2c = & 1d = −
4 8x − =
Important Idea
The following statements
mean the same thing:
The distance from x to 4 is 8
units.
Example
Solve 4 8x − =
Example
Solve 2 4 8x − =
Confirm your answer(s)
with your calculator.
Try This
4 2 8x − =Solve
5 3
,
2 2
x = −
Confirm with your calculator.
Try This
4 2 8x − = −
Solve
Confirm with your calculator.
?
Example
2
4 3 2x x+ − =
Solve using the algebraic
definition of absolute value:
Definition
Radical equations are
equations that contain
expressions under a radical
symbol.
Example
5 3 11x x+ − =
Solve:
Important Idea
It is always a good idea to
verify solutions (algebraically
or graphically). Solutions to
radical equations...
Try This
Solve. Check for
extraneous roots:
3 2 7x − =
17x =
Definition
The Power Principle is the
process of raising both sides
of an equation to the same
power. It is what we have
b...
Example
Solve algebraically. Be sure
to check for extraneous
roots.
2 3 7 2x x− − + =
Try This
Solve algebraically. Be sure
to check for extraneous
roots.
2 2 3 1x x+ − − =
x=2
Definition
Fractional Equations,
sometimes called Rational
Equations, are equations in
the form:
( )
0
( )
f x
g x
=
where...
Important Idea
To solve fractional equations
of the form:
( )
0
( )
f x
g x
=
find all solutions where
( ) 0g x ≠( ) 0f x ...
Example
Solve. Check for
extraneous roots.
2
2
6 1
0
2 9 5
x x
x x
− −
=
+ −
Try This
Solve. Check for
extraneous roots.
2
2
2 3
0
2 5 3
x x
x x
+ −
=
+ +
1x =
Lesson Close
The ability to solve all
types of equations is one of
the most basic skills in
mathematics.
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  • X=-2,6
  • X=-2,6
  • X=-2,6
  • X=4 (x=9 is extraneous)
  • X=42
  • X=-1/3 (3x+1)(2x-1)=0
  • Hprec2 4

    1. 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. 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. 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. 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. 5. Definition If c and d are real numbers, then is the distance between c and d on the number line. c d−
    6. 6. Example 0 10-10 Find the distance between 10 and -10: 10 ( 10) 10 10 20 20d = − − = + = = 20
    7. 7. Example 0 10-10 Find the distance between -10 and 10: 10 (10) 10 10 20 20d = − − = − − = − = 20
    8. 8. Try This True of False? c d c d+ = + Give an example to support your answer. False; 2c = & 1d = −
    9. 9. 4 8x − = Important Idea The following statements mean the same thing: The distance from x to 4 is 8 units.
    10. 10. Example Solve 4 8x − =
    11. 11. Example Solve 2 4 8x − = Confirm your answer(s) with your calculator.
    12. 12. Try This 4 2 8x − =Solve 5 3 , 2 2 x = − Confirm with your calculator.
    13. 13. Try This 4 2 8x − = − Solve Confirm with your calculator. ?
    14. 14. Example 2 4 3 2x x+ − = Solve using the algebraic definition of absolute value:
    15. 15. Definition Radical equations are equations that contain expressions under a radical symbol.
    16. 16. Example 5 3 11x x+ − = Solve:
    17. 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. 18. Try This Solve. Check for extraneous roots: 3 2 7x − = 17x =
    19. 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. 20. Example Solve algebraically. Be sure to check for extraneous roots. 2 3 7 2x x− − + =
    21. 21. Try This Solve algebraically. Be sure to check for extraneous roots. 2 2 3 1x x+ − − = x=2
    22. 22. Definition Fractional Equations, sometimes called Rational Equations, are equations in the form: ( ) 0 ( ) f x g x = where ( ) 0g x ≠
    23. 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. 24. Example Solve. Check for extraneous roots. 2 2 6 1 0 2 9 5 x x x x − − = + −
    25. 25. Try This Solve. Check for extraneous roots. 2 2 2 3 0 2 5 3 x x x x + − = + + 1x =
    26. 26. Lesson Close The ability to solve all types of equations is one of the most basic skills in mathematics.
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