Hprec2 4

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

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

DefinitionThe absolute value of a number c is denoted and is defined as follows:

c

If , then 0c c c

If , then 0c c c

ExampleIf , then 0c c c

If , then 0c c cUsing the definition of absolute value above, find the absolute value when:

2c 3c 0c

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 ThisTrue of False?

c d c d

Give an example to support your answer.

False;

2c & 1d

4 8x

Important IdeaThe 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 This4 2 8x Solve

5 3,2 2

x

Confirm with your calculator.

Try This4 2 8x

Solve

Confirm with your calculator.

?

Example

2 4 3 2x x

Solve using the algebraic definition of absolute value:

DefinitionRadical equations are equations that contain expressions under a radical symbol.

Example

5 3 11x x

Solve:

Important IdeaIt 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.

Try ThisSolve. Check for extraneous roots:

3 2 7x

17x

DefinitionThe 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.

ExampleSolve algebraically. Be sure to check for extraneous roots.

2 3 7 2x x

Try ThisSolve 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 ( ) 0g x

Important Idea To solve fractional equations of the form:

( )0

( )

f x

g x

find all solutions where ( ) 0g x ( ) 0f x and

Example Solve. Check for extraneous roots.

2

2

6 10

2 9 5

x x

x x

Try ThisSolve. Check for extraneous roots.

2

2

2 30

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.