1. Chapter 2: Equations and Inequalities 2.4: Other Types of Equations
Essential Question: How many solutions should you expect in an absolute value equation? A radical equation? A fractional equation?

2. 2.4: Other Types of Equations
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 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

3. 2.4: Other Types of Equations
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 or

4. 2.4: Other Types of Equations
Ex. 3: Solving an Absolute Value Quadratic Equation
Solve |x2 + 4x – 3| = 2 or

5. 2.4: Other Types of Equations
Page 116
9-21, all problems

6. Chapter 2: Equations and Inequalities 2
Chapter 2: Equations and Inequalities 2.4: Other Types of Equations Day 2
Essential Question: How many solutions should you expect in an absolute value equation? A radical equation? A fractional equation?

7. 2.4: Other Types of Equations
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)
isolate the radical
take each side to the inverted power e.g. square root → square both sides e.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

8. 2.4 Other Types of Equations
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

9. 2.4: Other Types of Equations
Sometimes the power principle must be applied twice
Ex. 5: Solve

10. 2.4: Other Types of Equations
Ex 5 (continued), 2nd application
Square both sides
FOIL left square each on right
Distribute
Get one side = 0
Factor
√ extraneous solutions

11. 2.4: Other Types of Equations
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:
Solve the numerator
Plug all answers in the denominator to avoid extraneous roots

12. 2.4: Other Types of Equations
Ex. 7: Solving a Fractional Equation
Solve
Find all solutions to 6x2 – x – 1 = 0

13. 2.4: Other Types of Equations
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

14. 2.4: Other Types of Equations
Assignment
Page 116 – 117
29 – 41 & 49 – 63
Odd problems (show work)