Essential Question: How many solutions should you expect in an absolute value equation? A

radical equation? A fractional equation?

Solving Absolute Value Equations◦ Get the absolute value term alone on one side of the

equation e.g. If you have 3|2x + 5| - 12 = 0,

add 12 to both sides of the equation,then divide both sides by 3 to get |2x + 5| = 4

◦ Create two equations and solve for x One positive (like the normal equation, without the | | signs) One negative (flip signs for all terms not inside the | |)

2x + 5 = 4 2x + 5 = -4◦ Check your answers for extraneous solutions

Solving Absolute Value Equalities◦ Ex. 2: Using the Algebraic Definition

Just like quadratic equations, where taking the square root of both sides left us with a positive or negative solution, removing absolute value requires us to solve for a positive and negative solution.

|x + 4| = 5x – 2 or4 5 2

5 6

4 6

3

2

x x

x x

x

x

4 (5 2)

4 5 2

5 2

6 2

2(extr

1aneous

3)

6

x x

x x

x x

x

x

Ex. 3: Solving an Absolute Value Quadratic Equation◦ Solve |x2 + 4x – 3| = 2◦ or2

2

4 3 2

4 5 0

( 5)( 1) 0

5 or 1

x x

x x

x x

x x

2

2

2

4 3 2

4 1 0

(4) (4) 4(1)( 1)

2(1)

4 16 4 4 20

2 2

4 4 5 4 2 5

2 2

2 5

x x

x x

x

x

x

x

Page 116 9-21, all problems

Essential Question: How many solutions should you expect in an absolute value equation? A

radical equation? A fractional equation?

Solving Radical Equations◦ Radical equations are equations that use a radical

(root) symbol. Graphing radical equations will only generate approximate solutions. Exact solutions need to be found algebraically.

◦ To remove a radical (Power principle)1. isolate the radical2. take each side to the inverted power

e.g. square root → square both sidese.g. cube root → cube both sides)

◦ Squaring both sides of an equation may introduce extraneous solutions, so solutions to radical equations MUST be checked in the original equation

Ex. 4: Solving a Radical Equation◦ Solve

isolate the radical

square both sides

FOIL the right

Get equation =0

Factor x = 9 or x = 4 √ Solutions

2

2

2

2

2

5 3 11

3 11 5

3 11 5

3 11 10 25

3

0 13 36

0 ( 9)

11 3 1

4

1

(

1

)

x

x x

x x

x x

x x x

x x

x x

xx

Sometimes the power principle must be applied twice◦ Ex. 5: Solve 2 3 7 2x x

2 2

2 3 7 2x x

2 3 7 2 2 7 4x x x

2 3 11 4 7x x x

14 4 7x x

Ex 5 (continued), 2nd application◦

Square both sides

FOIL leftsquare each on right

Distribute

Get one side = 0

Factor

√ extraneous solutions

14 4 7x x

2214 4 7x x

22 228 196 4 7x x x 2 28 196 16( 7)x x x 2 28 196 16 112x x x

2 44 84 0x x

( 2)( 42) 0x x

or 22 4x x

Fractional Equations◦ If f(x) and g(x) are algebraic expressions, the

quotient is called a fractional expression with

numerator f(x) and denominator g(x). As in all fractions, the denominator, g(x), cannot be zero.

◦ That is, if g(x) = 0, the fraction is undefined.◦ To solve a fractional equation:

1. Solve the numerator2. Plug all answers in the denominator to avoid

extraneous roots

( )

( )

f x

g x

Ex. 7: Solving a Fractional Equation◦ Solve

◦ Find all solutions to 6x2 – x – 1 = 0◦

2

2

6 10

2 9 5

x x

x x

26 1 0

(3 1)(2 1) 0

3 1 0 2 1 0

3 1 2 1

1 1

3 2

x x

x x

x or x

x or x

x or x

Check your solutions of x=½ and x=-⅓◦ Plug your answers from the numerator into the

denominator (2x2 + 9x – 5)

◦ -⅓ is a solution, and ½ is extraneous

2

2

1 1 702 9 5

3 3 9

1 12 9 5 0

2 2

Assignment◦ Page 116 – 117

29 – 41 & 49 – 63 Odd problems (show work)