Chapter 2: Equations and Inequalities 2.4: Other Types of Equations

1. Chapter 2: Equations and Inequalities2.4: 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 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

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 Inequalities2.4: Other Types of EquationsDay 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 powere.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

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)