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Comprehensive Guide to Solving Equations and Inequalities Using Various Methods

This detailed resource covers essential techniques for solving equations and inequalities, including multiplying by the least common multiple (LCM), applying the quadratic formula, and using the zero product property. Readers will learn how to simplify, check solutions, and analyze equations for real and imaginary roots. Various examples are provided, alongside methods like completing the square and synthetic division to find roots. This guide is perfect for students looking to strengthen their algebra skills and enhance their problem-solving strategies in math.

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Comprehensive Guide to Solving Equations and Inequalities Using Various Methods

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  1. Review 4.6-4.7

  2. Solve each equation or inequality 1. 2. Multiply every term by 12 b = 2, -5

  3. Solve each equation or inequality 3. 4. USE QUADRATIC FORMULA w = 2/5, -1 What values go on your number lines?? 2/5, -1, 0, and 1

  4. Example: Solve. LCM: 2x Multiply each fraction through by the LCM Check your solution!

  5. Solve. LCM: ? LCM: (x+1) ? Check your solution! No Solution!

  6. Solve. Factor 1st! LCM: (x+2)(x-2) Check your solutions!

  7. Example: Solve. Check your solutions!

  8. Last Example: Solve. Check your solutions!

  9. Solve Check your solution. The LCD for the three denominators is Original equation Multiply each sideby 24(3 – x). Example 6-1a

  10. 6 1 1 1 1 1 Simplify. Simplify. Add. Example 6-1b

  11. Check Original equation Simplify. Simplify. Example 6-1c The solution is correct.

  12. Answer: The solution is –45. Example 6-1d

  13. Solve Answer: Example 6-1e

  14. Solve Check your solution. The LCD is Original equation p – 1 1 Multiply by the LCD, (p2 – 1). 1 1 Example 6-2a

  15. DistributiveProperty Simplify. Simplify. Add(2p2 – 2p + 1)toeach side. Example 6-2b

  16. Divide eachside by 3. Factor. Zero ProductProperty or Solve eachequation. Example 6-2c

  17. Check Original equation Simplify. Simplify. Example 6-2d

  18. Original equation Simplify. Example 6-2e Since p = –1 results in a zero in the denominator, eliminate –1. Answer: The solution is p = 2.

  19. Solve Answer: Example 6-2f

  20. Decompose this into partial fractions 5.

  21. Solve Related equation Example 6-5a Step 1 Values that make the denominator equal to 0 are excluded from the denominator.For this inequality the excluded value is 0. Step 2 Solve the related equation.

  22. Multiply each side by 9s. Simplify. Add. Divide each side by 6. Example 6-5b

  23. Step 3 Draw vertical lines at the excluded value and at the solution to separate the number line into regions. Example 6-5c Now test a sample value in each region to determine if the values in the region satisfy the inequality.

  24. Test is a solution. Example 6-5d

  25. Test is not a solution. Example 6-5e

  26. Test is a solution. Example 6-5f

  27. Answer: The solution Example 6-5g

  28. Solve Answer: Example 6-5h

  29. Solve each equation or inequality 6. 7. x = 7 No real solution

  30. Solve each equation or inequality 8. 9. k = 7 m = -13

  31. Solve each equation or inequality 10. t > 14

  32. 11.) Find the upper and lower bound of the zeros of: Upper = 4 Lower = -1

  33. Approximate the real zeros of each functions 12. 13. Between 1 and 2 Between -1 and 0 Between 2 and 3

  34. Approximate the real zeros of each functions 14. 15. Between -5 and -4, -1 and 0, 1 and 2 No real zeros

  35. 16.) Find the number of positive, negative, and imaginary: Pos. 2 or 0 Neg. 1

  36. 17.) Find the possible rational roots:

  37. Find the remainder of each division. Then state whether the binomial is a factor? 18. 19. Remainder = 55 Not a factor Remainder = 18 Not a factor

  38. 20.) Use synthetic division to divide:

  39. Graph each equation or inequality: 21. 22.

  40. Solve each equation or inequality 23. 24.

  41. Solve each equation by completing the square: 25. 26.

  42. 27.) Find the discriminant of the function given and describe the nature of the roots. Discriminant = 73 2 distinct real roots

  43. Write the polynomial equation of least degree have the roots: 28.)1, -1, 0.5

  44. Find the roots of each equation: 29. 30.

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