1. 1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 1 Equations and Inequalities

2. OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Other Types of Equations Learn to solve equations by factoring. Learn to solve fractional equations. Learn to solve equations involving radicals. Learn to solve equations that are quadratic in form. SECTION 1.5 1 2 3 4 2

3. PROCEDURE FOR SOLVING EQUATIONS BY FACTORING Step 1Make one side zero. Move all nonzero terms in the equation to one side (say the left side), so that the other side (right side) is 0. Step 2Factor the left side. Step 3Use the zero-product property. Set each factor in Step 2 equal to 0, and then solve the resulting equations. Step 4Check your solutions. © 2010 Pearson Education, Inc. All rights reserved 3

4. EXAMPLE 2 Solving an Equation by Factoring Solve by factoring: The solution set is {–3, 0, 3}. Solution Step 1 Step 2 Step 3 Step 4 © 2010 Pearson Education, Inc. All rights reserved 4

5. EXAMPLE 3 Solving an Equation by Factoring Solve by factoring: The solution set is {2,i,–i}. Solution Step 1 Step 2 Step 3 Step 4 © 2010 Pearson Education, Inc. All rights reserved 5

6. EXAMPLE 4 Solving a Rational Equation Solve: Solution Step 1Find the LCD: 6x(x + 1) Step 2 © 2010 Pearson Education, Inc. All rights reserved 6

7. The solution set is {–3,2}. Solution continued Step 4 Step 3 Step 5Both solutions check in the original equation. © 2010 Pearson Education, Inc. All rights reserved 7 EXAMPLE 4 Solving a Rational Equation

8. EXAMPLE 5 Solving a Rational Equation with an Extraneous Solution Solve: Solution Step 1LCD: (x – 1)(x + 1) = x 2 + 1 Step 2 © 2010 Pearson Education, Inc. All rights reserved 8

9. Solution continued © 2010 Pearson Education, Inc. All rights reserved 9 EXAMPLE 5 Solving a Rational Equation with an Extraneous Solution

10. Solution continued Step 4 x – 1 = 0 or x = 1 Step 3 (x – 1)(x – 1) = 0 Step 5 Check: © 2010 Pearson Education, Inc. All rights reserved 10 EXAMPLE 5 Solving a Rational Equation with an Extraneous Solution The denominator of the first term is 0, so the equation is undefined for x = 1. Therefore, the original equation has no solution.

11. EXAMPLE 6 Solving Equations Involving Radicals Solve: Solution Since we raise both sides to power 2. © 2010 Pearson Education, Inc. All rights reserved 11

12. EXAMPLE 6 Solving Equations Involving Radicals Solution continued –3 is an extraneous solution. The solution set is {0, 2}. Check each solution. ? ? ? © 2010 Pearson Education, Inc. All rights reserved 12

13. EXAMPLE 7 Solving an Equations Involving a Radical Solve: Solution Step 1Isolate the radical on one side. Step 2Square both sides and simplify. © 2010 Pearson Education, Inc. All rights reserved 13

14. Solution continued Step 3Set each factor = 0. 0 is an extraneous solution. The solution set is {4}. Step 4Check. ? ? © 2010 Pearson Education, Inc. All rights reserved 14 EXAMPLE 7 Solving an Equations Involving a Radical

15. EXAMPLE 8 Solving an Equation Involving Two Radicals Solve: Solution Step 1Isolate one of the radicals. Step 2Square both sides and simplify. © 2010 Pearson Education, Inc. All rights reserved 15

16. Solution continued Step 3Repeat the process - isolate the radical, square both sides, simplify and factor. © 2010 Pearson Education, Inc. All rights reserved 16 EXAMPLE 8 Solving an Equation Involving Two Radicals

17. Solution continued Step 4Set each factor = 0. The solution set is {1,5}. Step 5Check. © 2010 Pearson Education, Inc. All rights reserved 17 EXAMPLE 8 Solving an Equation Involving Two Radicals

18. SOLVING EQUATIONS CONTAINING SQUARE ROOTS Step 1Isolate one radical to one side of the equation. Step 2Square both sides of the equation in Step 1 and simplify. Step 3If the equation in Step 2 contains a radical, repeat Steps 1 and 2 to get an equation that is free of radicals. Step 5Check the solutions in the original equation. Step 4Solve the equation obtained in Steps 1 - 3. © 2010 Pearson Education, Inc. All rights reserved 18

19. SOLVING EQUATIONS OF THE FORM u m/n = k © 2010 Pearson Education, Inc. All rights reserved 19 Let m and n be positive integers, k a real number, and in lowest terms. Then if

20. EXAMPLE 9 Solving Equations with Rational Exponents Solve. Solution © 2010 Pearson Education, Inc. All rights reserved 20

21. EXAMPLE 9 Solving Equations with Rational Exponents Solution continued © 2010 Pearson Education, Inc. All rights reserved 21 Check:

22. EXAMPLE 9 Solving Equations with Rational Exponents Solution continued © 2010 Pearson Education, Inc. All rights reserved 22 Then

23. EXAMPLE 9 Solving Equations with Rational Exponents Solution continued © 2010 Pearson Education, Inc. All rights reserved 23 Both x = 3 and satisfy the original equation. So the solution set is

24. An equation in a variable x is quadratic in form if it can be written as EQUATIONS THAT ARE QUADRATIC IN FORM where u is an expression in the variable x. We solve the equation au 2 + bu + c = 0 for u. Then the solutions of the original equation can be obtained by replacing u by the expression in x that u represents. © 2010 Pearson Education, Inc. All rights reserved 24

25. EXAMPLE 10 Solving an Equation That Is Quadratic in Form Solve: Solution Let u = x 1/3, then u 2 = (x 1/3 ) 2 = x 2/3. © 2010 Pearson Education, Inc. All rights reserved 25

26. EXAMPLE 10 Solving an Equation That Is Quadratic in Form Solution continued Replace u with x 1/3. © 2010 Pearson Education, Inc. All rights reserved 26 Find x from the equations u = 2 and u = 3, Checking shows that both x = 8 and x = 27 satisfy the equation. The solution set is {8, 27}.

27. EXAMPLE 11 Solving an Equation That Is Quadratic in Form Solve: Solution Let then © 2010 Pearson Education, Inc. All rights reserved 27

28. EXAMPLE 11 Solving an Equation That Is Quadratic in Form Solution continued Replace u with and solve for x. x = 1 checks in the original equation © 2010 Pearson Education, Inc. All rights reserved 28

29. EXAMPLE 11 Solving an Equation That Is Quadratic in Form Solution continued Both solutions check in the original equation. The solution set is © 2010 Pearson Education, Inc. All rights reserved 29

30. EXAMPLE 12 Investigating Space Travel Your sister is 5 years older than you are. She decides she has had enough of Earth and needs a vacation. She takes a trip to the Omega-One star system. Her trip to Omega-One and back in a spacecraft traveling at an average speed v took 15 years, according to the clock and calendar on the spacecraft. But on landing back on Earth, she discovers that her voyage took 25 years, according to the time on Earth. © 2010 Pearson Education, Inc. All rights reserved 30

31. EXAMPLE 12 Investigating Space Travel This means that, although you were 5 years younger than your sister before her vacation, you are 5 years older than her after her vacation! Use the time-dilation equation to calculate the speed of the spacecraft. © 2010 Pearson Education, Inc. All rights reserved 31

32. EXAMPLE 12 Investigating Space Travel Substitute t 0 = 15 (moving-frame time) and t = 25 (fixed-frame time) to obtain Solution So the spacecraft was moving at 80% (0.8c) the speed of light. © 2010 Pearson Education, Inc. All rights reserved 32