1.4 sign charts and inequalities i

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1.4 sign charts and inequalities i

  1. 1. Inequalities http://www.lahc.edu/math/precalculus/math_260a.html
  2. 2. Sign-Charts and Inequalities I Given an expression f, it’s important to identify when the output is positive (f > 0) and when the output is negative (f < 0) when f is evaluated with a value x.
  3. 3. Example A. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2, -1/2. Sign-Charts and Inequalities I Given an expression f, it’s important to identify when the output is positive (f > 0) and when the output is negative (f < 0) when f is evaluated with a value x.
  4. 4. Sign-Charts and Inequalities I For polynomials or rational expressions, factor them to determine the signs of their outputs. Given an expression f, it’s important to identify when the output is positive (f > 0) and when the output is negative (f < 0) when f is evaluated with a value x. Example A. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2, -1/2.
  5. 5. In factored form x2 – 2x – 3 = (x – 3)(x + 1) Sign-Charts and Inequalities I For polynomials or rational expressions, factor them to determine the signs of their outputs. Given an expression f, it’s important to identify when the output is positive (f > 0) and when the output is negative (f < 0) when f is evaluated with a value x. Example A. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2, -1/2.
  6. 6. In factored form x2 – 2x – 3 = (x – 3)(x + 1) Hence, for x = -3/2: (-3/2 – 3)(-3/2 + 1) Sign-Charts and Inequalities I For polynomials or rational expressions, factor them to determine the signs of their outputs. Given an expression f, it’s important to identify when the output is positive (f > 0) and when the output is negative (f < 0) when f is evaluated with a value x. Example A. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2, -1/2.
  7. 7. In factored form x2 – 2x – 3 = (x – 3)(x + 1) Hence, for x = -3/2: (-3/2 – 3)(-3/2 + 1) is (–)(–) = + . Sign-Charts and Inequalities I For polynomials or rational expressions, factor them to determine the signs of their outputs. Given an expression f, it’s important to identify when the output is positive (f > 0) and when the output is negative (f < 0) when f is evaluated with a value x. Example A. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2, -1/2.
  8. 8. In factored form x2 – 2x – 3 = (x – 3)(x + 1) Hence, for x = -3/2: (-3/2 – 3)(-3/2 + 1) is (–)(–) = + . And for x = -1/2: (-1/2 – 3)(-1/2 + 1) Sign-Charts and Inequalities I For polynomials or rational expressions, factor them to determine the signs of their outputs. Given an expression f, it’s important to identify when the output is positive (f > 0) and when the output is negative (f < 0) when f is evaluated with a value x. Example A. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2, -1/2.
  9. 9. In factored form x2 – 2x – 3 = (x – 3)(x + 1) Hence, for x = -3/2: (-3/2 – 3)(-3/2 + 1) is (–)(–) = + . And for x = -1/2: (-1/2 – 3)(-1/2 + 1) is (–)(+) = – . Sign-Charts and Inequalities I For polynomials or rational expressions, factor them to determine the signs of their outputs. Given an expression f, it’s important to identify when the output is positive (f > 0) and when the output is negative (f < 0) when f is evaluated with a value x. Example A. Determine whether the outcome is + or – for x2 – 2x – 3 if x = -3/2, -1/2.
  10. 10. Sign-Charts and Inequalities I Example B. Determine whether the outcome is x2 – 2x – 3 x2 + x – 2 if x = -3/2, -1/2.+ or – for
  11. 11. Example B. Determine whether the outcome is x2 – 2x – 3 x2 + x – 2 x2 – 2x – 3 x2 + x – 2 In factored form = (x – 3)(x + 1) (x – 1)(x + 2) Sign-Charts and Inequalities I if x = -3/2, -1/2.+ or – for
  12. 12. x2 – 2x – 3 x2 + x – 2 In factored form = (x – 3)(x + 1) (x – 1)(x + 2) Hence, for x = -3/2: (x – 3)(x + 1) (x – 1)(x + 2) = (–)(–) (–)(+) Sign-Charts and Inequalities I Example B. Determine whether the outcome is x2 – 2x – 3 x2 + x – 2 if x = -3/2, -1/2.+ or – for
  13. 13. x2 – 2x – 3 x2 + x – 2 In factored form = (x – 3)(x + 1) (x – 1)(x + 2) Hence, for x = -3/2: (x – 3)(x + 1) (x – 1)(x + 2) = (–)(–) (–)(+) < 0 Sign-Charts and Inequalities I Example B. Determine whether the outcome is x2 – 2x – 3 x2 + x – 2 if x = -3/2, -1/2.+ or – for
  14. 14. x2 – 2x – 3 x2 + x – 2 In factored form = (x – 3)(x + 1) (x – 1)(x + 2) Hence, for x = -3/2: (x – 3)(x + 1) (x – 1)(x + 2) = (–)(–) (–)(+) < 0 For x = -1/2: (x – 3)(x + 1) (x – 1)(x + 2) = (–)(+) (–)(+) Sign-Charts and Inequalities I Example B. Determine whether the outcome is x2 – 2x – 3 x2 + x – 2 if x = -3/2, -1/2.+ or – for
  15. 15. x2 – 2x – 3 x2 + x – 2 In factored form = (x – 3)(x + 1) (x – 1)(x + 2) Hence, for x = -3/2: (x – 3)(x + 1) (x – 1)(x + 2) = (–)(–) (–)(+) < 0 For x = -1/2: (x – 3)(x + 1) (x – 1)(x + 2) = (–)(+) (–)(+) > 0 Sign-Charts and Inequalities I Example B. Determine whether the outcome is x2 – 2x – 3 x2 + x – 2 if x = -3/2, -1/2.+ or – for
  16. 16. x2 – 2x – 3 x2 + x – 2 In factored form = (x – 3)(x + 1) (x – 1)(x + 2) Hence, for x = -3/2: (x – 3)(x + 1) (x – 1)(x + 2) = (–)(–) (–)(+) < 0 For x = -1/2: (x – 3)(x + 1) (x – 1)(x + 2) = (–)(+) (–)(+) > 0 This leads to the sign charts of formulas. The sign- chart of a formula gives the signs of the outputs. Sign-Charts and Inequalities I Example B. Determine whether the outcome is x2 – 2x – 3 x2 + x – 2 if x = -3/2, -1/2.+ or – for
  17. 17. Here is an example, the sign chart of f = x – 1: 1 f = 0 + +– – – – x – 1 Sign-Charts and Inequalities I
  18. 18. Here is an example, the sign chart of f = x – 1: 1 f = 0 + +– – – – x – 1 The "+" indicates the region where the output is positive i.e. if 1 < x. Sign-Charts and Inequalities I
  19. 19. Here is an example, the sign chart of f = x – 1: 1 f = 0 + +– – – – x – 1 The "+" indicates the region where the output is positive i.e. if 1 < x. Likewise, the "–" indicates the region where the output is negative, i.e. x < 1. Sign-Charts and Inequalities I
  20. 20. Construction of the sign-chart of f. Here is an example, the sign chart of f = x – 1: 1 f = 0 + +– – – – x – 1 The "+" indicates the region where the output is positive i.e. if 1 < x. Likewise, the "–" indicates the region where the output is negative, i.e. x < 1. Sign-Charts and Inequalities I
  21. 21. Construction of the sign-chart of f. I. Solve for f = 0 (and denominator = 0) if there is any denominator. Here is an example, the sign chart of f = x – 1: 1 f = 0 + +– – – – x – 1 The "+" indicates the region where the output is positive i.e. if 1 < x. Likewise, the "–" indicates the region where the output is negative, i.e. x < 1. Sign-Charts and Inequalities I
  22. 22. Construction of the sign-chart of f. I. Solve for f = 0 (and denominator = 0) if there is any denominator. II. Draw the real line, mark off the answers from I. Here is an example, the sign chart of f = x – 1: 1 f = 0 + +– – – – x – 1 The "+" indicates the region where the output is positive i.e. if 1 < x. Likewise, the "–" indicates the region where the output is negative, i.e. x < 1. Sign-Charts and Inequalities I
  23. 23. Construction of the sign-chart of f. I. Solve for f = 0 (and denominator = 0) if there is any denominator. II. Draw the real line, mark off the answers from I. III. Sample each segment for signs by testing a point in each segment. Here is an example, the sign chart of f = x – 1: 1 f = 0 + +– – – – x – 1 The "+" indicates the region where the output is positive i.e. if 1 < x. Likewise, the "–" indicates the region where the output is negative, i.e. x < 1. Sign-Charts and Inequalities I
  24. 24. Construction of the sign-chart of f. I. Solve for f = 0 (and denominator = 0) if there is any denominator. II. Draw the real line, mark off the answers from I. III. Sample each segment for signs by testing a point in each segment. Here is an example, the sign chart of f = x – 1: 1 f = 0 + +– – – – x – 1 The "+" indicates the region where the output is positive i.e. if 1 < x. Likewise, the "–" indicates the region where the output is negative, i.e. x < 1. Fact: The sign stays the same for x's in between the values from step I (where f = 0 or f is undefined.) Sign-Charts and Inequalities I
  25. 25. Example C. Let f = x2 – 3x – 4 , use the sign- chart to indicate when is f = 0, f > 0, and f < 0. Sign-Charts and Inequalities I
  26. 26. Example C. Let f = x2 – 3x – 4 , use the sign- chart to indicate when is f = 0, f > 0, and f < 0. Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0  x = 4 , -1 Sign-Charts and Inequalities I
  27. 27. Example C. Let f = x2 – 3x – 4 , use the sign- chart to indicate when is f = 0, f > 0, and f < 0. Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0  x = 4 , -1 Mark off these points on a line: (x-4)(x+1) 4-1 Sign-Charts and Inequalities I
  28. 28. Example C. Let f = x2 – 3x – 4 , use the sign- chart to indicate when is f = 0, f > 0, and f < 0. Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0  x = 4 , -1 Mark off these points on a line: (x-4)(x+1) Select points to sample in each segment: 4-1 Sign-Charts and Inequalities I
  29. 29. Example C. Let f = x2 – 3x – 4 , use the sign- chart to indicate when is f = 0, f > 0, and f < 0. Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0  x = 4 , -1 Mark off these points on a line: (x-4)(x+1) 4-1 Select points to sample in each segment: Test x = - 2, -2 Sign-Charts and Inequalities I
  30. 30. Example C. Let f = x2 – 3x – 4 , use the sign- chart to indicate when is f = 0, f > 0, and f < 0. Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0  x = 4 , -1 Mark off these points on a line: (x-4)(x+1) 4-1 Select points to sample in each segment: Test x = - 2, get – * – = + . Hence the segment is positive. Draw + sign over it. -2 Sign-Charts and Inequalities I
  31. 31. Example C. Let f = x2 – 3x – 4 , use the sign- chart to indicate when is f = 0, f > 0, and f < 0. Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0  x = 4 , -1 Mark off these points on a line: (x-4)(x+1) + + + + + 4-1 Select points to sample in each segment: Test x = - 2, get – * – = + . Hence the segment is positive. Draw + sign over it. -2 Sign-Charts and Inequalities I
  32. 32. Example C. Let f = x2 – 3x – 4 , use the sign- chart to indicate when is f = 0, f > 0, and f < 0. Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0  x = 4 , -1 Mark off these points on a line: (x-4)(x+1) + + + + + 0 4-1 Select points to sample in each segment: Test x = - 2, get – * – = + . Hence the segment is positive. Draw + sign over it. -2 Test x = 0, get – * + = –. Hence this segment is negative. Put – over it. Sign-Charts and Inequalities I
  33. 33. Example C. Let f = x2 – 3x – 4 , use the sign- chart to indicate when is f = 0, f > 0, and f < 0. Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0  x = 4 , -1 Mark off these points on a line: (x-4)(x+1) + + + + + – – – – – 0 4-1 Select points to sample in each segment: Test x = - 2, get – * – = + . Hence the segment is positive. Draw + sign over it. -2 Test x = 0, get – * + = –. Hence this segment is negative. Put – over it. Sign-Charts and Inequalities I
  34. 34. Example C. Let f = x2 – 3x – 4 , use the sign- chart to indicate when is f = 0, f > 0, and f < 0. Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0  x = 4 , -1 Mark off these points on a line: (x-4)(x+1) + + + + + – – – – – 0 4-1 Select points to sample in each segment: Test x = - 2, get – * – = + . Hence the segment is positive. Draw + sign over it. -2 Test x = 0, get – * + = –. Hence this segment is negative. Put – over it. Test x = 5, get + * + = +. Hence this segment is positive. Put + over it. 5 Sign-Charts and Inequalities I
  35. 35. Example C. Let f = x2 – 3x – 4 , use the sign- chart to indicate when is f = 0, f > 0, and f < 0. Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0  x = 4 , -1 Mark off these points on a line: (x-4)(x+1) + + + + + – – – – – + + + + + 0 4-1 Select points to sample in each segment: Test x = - 2, get – * – = + . Hence the segment is positive. Draw + sign over it. -2 Test x = 0, get – * + = –. Hence this segment is negative. Put – over it. Test x = 5, get + * + = +. Hence this segment is positive. Put + over it. 5 Sign-Charts and Inequalities I
  36. 36. Example D. Make the sign chart of f = (x – 3) (x – 1)(x + 2) Sign-Charts and Inequalities I
  37. 37. Example D. Make the sign chart of f = (x – 3) (x – 1)(x + 2) The root for f = 0 is from the zero of the numerator which is x = 3. Sign-Charts and Inequalities I
  38. 38. Example D. Make the sign chart of f = (x – 3) (x – 1)(x + 2) The root for f = 0 is from the zero of the numerator which is x = 3. The zeroes of the denominator x = 1, -2 are the values where f is undefined (UDF). Sign-Charts and Inequalities I
  39. 39. Example D. Make the sign chart of f = (x – 3) (x – 1)(x + 2) The root for f = 0 is from the zero of the numerator which is x = 3. The zeroes of the denominator x = 1, -2 are the values where f is undefined (UDF). Mark these values on a real line. Sign-Charts and Inequalities I
  40. 40. Example D. Make the sign chart of f = (x – 3) (x – 1)(x + 2) The root for f = 0 is from the zero of the numerator which is x = 3. The zeroes of the denominator x = 1, -2 are the values where f is undefined (UDF). Mark these values on a real line. (x – 3) (x – 1)(x + 2) -2 1 3 UDF UDF f=0 Sign-Charts and Inequalities I
  41. 41. Example D. Make the sign chart of f = Select a point to sample in each segment: (x – 3) (x – 1)(x + 2) The root for f = 0 is from the zero of the numerator which is x = 3. The zeroes of the denominator x = 1, -2 are the values where f is undefined (UDF). Mark these values on a real line. (x – 3) (x – 1)(x + 2) -2 1 3 UDF UDF f=0 -3 0 2 4 Sign-Charts and Inequalities I
  42. 42. Example D. Make the sign chart of f = Select a point to sample in each segment: Test x = -3, we've a (x – 3) (x – 1)(x + 2) The root for f = 0 is from the zero of the numerator which is x = 3. The zeroes of the denominator x = 1, -2 are the values where f is undefined (UDF). Mark these values on a real line. (x – 3) (x – 1)(x + 2) -2 1 3 UDF UDF f=0 -3 ( – ) ( – )( – ) = – segment. 0 2 4 Sign-Charts and Inequalities I
  43. 43. Example D. Make the sign chart of f = Select a point to sample in each segment: Test x = -3, we've a (x – 3) (x – 1)(x + 2) The root for f = 0 is from the zero of the numerator which is x = 3. The zeroes of the denominator x = 1, -2 are the values where f is undefined (UDF). Mark these values on a real line. (x – 3) (x – 1)(x + 2) -2 1 3 UDF UDF f=0 -3 ( – ) ( – )( – ) = – segment. 0 2 4 Test x = 0, we've a ( – ) ( – )( + ) = + segment. Sign-Charts and Inequalities I
  44. 44. Example D. Make the sign chart of f = Select a point to sample in each segment: Test x = -3, we've a (x – 3) (x – 1)(x + 2) The root for f = 0 is from the zero of the numerator which is x = 3. The zeroes of the denominator x = 1, -2 are the values where f is undefined (UDF). Mark these values on a real line. (x – 3) (x – 1)(x + 2) -2 1 3 UDF UDF f=0 -3 ( – ) ( – )( – ) = – segment. 0 2 4 Test x = 0, we've a ( – ) ( – )( + ) = + segment. Test x = 2, we've a ( – ) ( + )( + ) segment. = – Sign-Charts and Inequalities I
  45. 45. Example D. Make the sign chart of f = Select a point to sample in each segment: Test x = -3, we've a (x – 3) (x – 1)(x + 2) The root for f = 0 is from the zero of the numerator which is x = 3. The zeroes of the denominator x = 1, -2 are the values where f is undefined (UDF). Mark these values on a real line. (x – 3) (x – 1)(x + 2) -2 1 3 UDF UDF f=0 -3 ( – ) ( – )( – ) = – segment. 0 2 4 Test x = 0, we've a ( – ) ( – )( + ) = + segment. Test x = 2, we've a ( – ) ( + )( + ) segment. = – Test x = 4, we've a ( + ) ( + )( + ) segment. = + – – – – + + + – – – + + + + Sign-Charts and Inequalities I
  46. 46. The easiest way to solve a polynomial or rational inequality is to use the sign-chart. Sign-Charts and Inequalities I
  47. 47. Example E. Solve x2 – 3x > 4 The easiest way to solve a polynomial or rational inequality is to use the sign-chart. Sign-Charts and Inequalities I
  48. 48. Example E. Solve x2 – 3x > 4 The easiest way to solve a polynomial or rational inequality is to use the sign-chart. To do this, I. set one side of the inequality to 0, Sign-Charts and Inequalities I
  49. 49. Example E. Solve x2 – 3x > 4 The easiest way to solve a polynomial or rational inequality is to use the sign-chart. To do this, I. set one side of the inequality to 0, Setting one side to 0, we have x2 – 3x – 4 > 0 Sign-Charts and Inequalities I
  50. 50. Example E. Solve x2 – 3x > 4 The easiest way to solve a polynomial or rational inequality is to use the sign-chart. To do this, I. set one side of the inequality to 0, II. factor the expression Setting one side to 0, we have x2 – 3x – 4 > 0 Sign-Charts and Inequalities I
  51. 51. Example E. Solve x2 – 3x > 4 The easiest way to solve a polynomial or rational inequality is to use the sign-chart. To do this, I. set one side of the inequality to 0, II. factor the expression Setting one side to 0, we have x2 – 3x – 4 > 0 or (x – 4)(x + 1) > 0. Sign-Charts and Inequalities I
  52. 52. Example E. Solve x2 – 3x > 4 The easiest way to solve a polynomial or rational inequality is to use the sign-chart. To do this, I. set one side of the inequality to 0, II. factor the expression and draw the sign-chart, Setting one side to 0, we have x2 – 3x – 4 > 0 or (x – 4)(x + 1) > 0. Sign-Charts and Inequalities I
  53. 53. Example E. Solve x2 – 3x > 4 The easiest way to solve a polynomial or rational inequality is to use the sign-chart. To do this, I. set one side of the inequality to 0, II. factor the expression and draw the sign-chart, Setting one side to 0, we have x2 – 3x – 4 > 0 or (x – 4)(x + 1) > 0. The roots are x = -1, 4. Sign-Charts and Inequalities I
  54. 54. Example E. Solve x2 – 3x > 4 4-1 The easiest way to solve a polynomial or rational inequality is to use the sign-chart. To do this, I. set one side of the inequality to 0, II. factor the expression and draw the sign-chart, Draw the sign-chart, sample the points x = -2, 0, 5 (x – 4)(x + 1) Setting one side to 0, we have x2 – 3x – 4 > 0 or (x – 4)(x + 1) > 0. The roots are x = -1, 4. Sign-Charts and Inequalities I
  55. 55. Example E. Solve x2 – 3x > 4 0 4-1-2 5 The easiest way to solve a polynomial or rational inequality is to use the sign-chart. To do this, I. set one side of the inequality to 0, II. factor the expression and draw the sign-chart, Draw the sign-chart, sample the points x = -2, 0, 5 (x – 4)(x + 1) Setting one side to 0, we have x2 – 3x – 4 > 0 or (x – 4)(x + 1) > 0. The roots are x = -1, 4. Sign-Charts and Inequalities I
  56. 56. Example E. Solve x2 – 3x > 4 0 4-1-2 5 The easiest way to solve a polynomial or rational inequality is to use the sign-chart. To do this, I. set one side of the inequality to 0, II. factor the expression and draw the sign-chart, Draw the sign-chart, sample the points x = -2, 0, 5 (x – 4)(x + 1) + + + Setting one side to 0, we have x2 – 3x – 4 > 0 or (x – 4)(x + 1) > 0. The roots are x = -1, 4. Sign-Charts and Inequalities I
  57. 57. Example E. Solve x2 – 3x > 4 0 4-1-2 5 The easiest way to solve a polynomial or rational inequality is to use the sign-chart. To do this, I. set one side of the inequality to 0, II. factor the expression and draw the sign-chart, Draw the sign-chart, sample the points x = -2, 0, 5 (x – 4)(x + 1) + + + – – – – – – Setting one side to 0, we have x2 – 3x – 4 > 0 or (x – 4)(x + 1) > 0. The roots are x = -1, 4. Sign-Charts and Inequalities I
  58. 58. Example E. Solve x2 – 3x > 4 0 4-1-2 5 The easiest way to solve a polynomial or rational inequality is to use the sign-chart. To do this, I. set one side of the inequality to 0, II. factor the expression and draw the sign-chart, Draw the sign-chart, sample the points x = -2, 0, 5 (x – 4)(x + 1) + + + – – – – – – + + + + Setting one side to 0, we have x2 – 3x – 4 > 0 or (x – 4)(x + 1) > 0. The roots are x = -1, 4. Sign-Charts and Inequalities I
  59. 59. Example E. Solve x2 – 3x > 4 0 4-1-2 5 The easiest way to solve a polynomial or rational inequality is to use the sign-chart. To do this, I. set one side of the inequality to 0, II. factor the expression and draw the sign-chart, III. read off the answer from the sign chart. Draw the sign-chart, sample the points x = -2, 0, 5 (x – 4)(x + 1) + + + – – – – – – + + + + Setting one side to 0, we have x2 – 3x – 4 > 0 or (x – 4)(x + 1) > 0. The roots are x = -1, 4. Sign-Charts and Inequalities I
  60. 60. Example E. Solve x2 – 3x > 4 0 4-1 The solutions are the + regions: -2 5 The easiest way to solve a polynomial or rational inequality is to use the sign-chart. To do this, I. set one side of the inequality to 0, II. factor the expression and draw the sign-chart, III. read off the answer from the sign chart. Draw the sign-chart, sample the points x = -2, 0, 5 (x – 4)(x + 1) + + + – – – – – – + + + + Setting one side to 0, we have x2 – 3x – 4 > 0 or (x – 4)(x + 1) > 0. The roots are x = -1, 4. Sign-Charts and Inequalities I
  61. 61. Example E. Solve x2 – 3x > 4 0 4-1 The solutions are the + regions: (–∞, –1) U (4, ∞) -2 5 4-1 The easiest way to solve a polynomial or rational inequality is to use the sign-chart. To do this, I. set one side of the inequality to 0, II. factor the expression and draw the sign-chart, III. read off the answer from the sign chart. Draw the sign-chart, sample the points x = -2, 0, 5 (x – 4)(x + 1) + + + – – – – – – + + + + Setting one side to 0, we have x2 – 3x – 4 > 0 or (x – 4)(x + 1) > 0. The roots are x = -1, 4. Sign-Charts and Inequalities I
  62. 62. Example E. Solve x2 – 3x > 4 0 4-1 The solutions are the + regions: (–∞, –1) U (4, ∞) -2 5 4-1 Note: The empty dot means those numbers are excluded. The easiest way to solve a polynomial or rational inequality is to use the sign-chart. To do this, I. set one side of the inequality to 0, II. factor the expression and draw the sign-chart, III. read off the answer from the sign chart. Draw the sign-chart, sample the points x = -2, 0, 5 (x – 4)(x + 1) + + + – – – – – – + + + + Setting one side to 0, we have x2 – 3x – 4 > 0 or (x – 4)(x + 1) > 0. The roots are x = -1, 4. Sign-Charts and Inequalities I
  63. 63. Example F. Solve x – 2 2 < x – 1 3 Sign-Charts and Inequalities I
  64. 64. Example F. Solve x – 2 2 < x – 1 3 Set the inequality to 0, x – 2 2 x – 1 3 < 0 Sign-Charts and Inequalities I
  65. 65. Example F. Solve x – 2 2 < x – 1 3 Set the inequality to 0, x – 2 2 x – 1 3 < 0 Put the expression into factored form, Sign-Charts and Inequalities I
  66. 66. Example F. Solve x – 2 2 < x – 1 3 Set the inequality to 0, x – 2 2 x – 1 3 < 0 Put the expression into factored form, x – 2 2 x – 1 3 = (x – 2)(x – 1) 2(x – 1) – 3(x – 2) Sign-Charts and Inequalities I
  67. 67. Example F. Solve x – 2 2 < x – 1 3 Set the inequality to 0, x – 2 2 x – 1 3 < 0 Put the expression into factored form, x – 2 2 x – 1 3 = (x – 2)(x – 1) 2(x – 1) – 3(x – 2) = (x – 2)(x – 1) – x + 4 Sign-Charts and Inequalities I
  68. 68. Example F. Solve x – 2 2 < x – 1 3 Set the inequality to 0, x – 2 2 x – 1 3 < 0 Put the expression into factored form, x – 2 2 x – 1 3 = (x – 2)(x – 1) 2(x – 1) – 3(x – 2) = (x – 2)(x – 1) – x + 4 Hence the inequality is (x – 2)(x – 1) – x + 4 < 0 Sign-Charts and Inequalities I
  69. 69. Example F. Solve x – 2 2 < x – 1 3 Set the inequality to 0, x – 2 2 x – 1 3 < 0 Put the expression into factored form, x – 2 2 x – 1 3 = (x – 2)(x – 1) 2(x – 1) – 3(x – 2) = (x – 2)(x – 1) – x + 4 Hence the inequality is (x – 2)(x – 1) – x + 4 < 0 It has a root at x = 4, and it’s undefined at x = 1, 2. Sign-Charts and Inequalities I
  70. 70. Example F. Solve x – 2 2 < x – 1 3 Set the inequality to 0, x – 2 2 x – 1 3 < 0 Put the expression into factored form, x – 2 2 x – 1 3 = (x – 2)(x – 1) 2(x – 1) – 3(x – 2) = (x – 2)(x – 1) – x + 4 Hence the inequality is (x – 2)(x – 1) – x + 4 < 0 Draw the sign chart by sampling x = 0, 3/2, 3, 5 It has a root at x = 4, and it's undefined at x = 1, 2. Sign-Charts and Inequalities I
  71. 71. Example F. Solve x – 2 2 < x – 1 3 Set the inequality to 0, x – 2 2 x – 1 3 < 0 Put the expression into factored form, x – 2 2 x – 1 3 = (x – 2)(x – 1) 2(x – 1) – 3(x – 2) = (x – 2)(x – 1) – x + 4 Hence the inequality is (x – 2)(x – 1) – x + 4 < 0 Draw the sign chart by sampling x = 0, 3/2, 3, 5 It has a root at x = 4, and it's undefined at x = 1, 2. 41 2 UDF UDF (x – 2)(x – 1) – x + 4 Sign-Charts and Inequalities I
  72. 72. Example F. Solve x – 2 2 < x – 1 3 Set the inequality to 0, x – 2 2 x – 1 3 < 0 Put the expression into factored form, x – 2 2 x – 1 3 = (x – 2)(x – 1) 2(x – 1) – 3(x – 2) = (x – 2)(x – 1) – x + 4 Hence the inequality is (x – 2)(x – 1) – x + 4 < 0 Draw the sign chart by sampling x = 0, 3/2, 3, 5 It has a root at x = 4, and it's undefined at x = 1, 2. 410 523/2 3 UDF UDF (x – 2)(x – 1) – x + 4 Sign-Charts and Inequalities I
  73. 73. Example F. Solve x – 2 2 < x – 1 3 Set the inequality to 0, x – 2 2 x – 1 3 < 0 Put the expression into factored form, x – 2 2 x – 1 3 = (x – 2)(x – 1) 2(x – 1) – 3(x – 2) = (x – 2)(x – 1) – x + 4 Hence the inequality is (x – 2)(x – 1) – x + 4 < 0 Draw the sign chart by sampling x = 0, 3/2, 3, 5 It has a root at x = 4, and it's undefined at x = 1, 2. 410 5 + + + 23/2 3 UDF UDF (x – 2)(x – 1) – x + 4 Sign-Charts and Inequalities I
  74. 74. Example F. Solve x – 2 2 < x – 1 3 Set the inequality to 0, x – 2 2 x – 1 3 < 0 Put the expression into factored form, x – 2 2 x – 1 3 = (x – 2)(x – 1) 2(x – 1) – 3(x – 2) = (x – 2)(x – 1) – x + 4 Hence the inequality is (x – 2)(x – 1) – x + 4 < 0 Draw the sign chart by sampling x = 0, 3/2, 3, 5 It has a root at x = 4, and it's undefined at x = 1, 2. 410 5 + + + – – 23/2 3 UDF UDF (x – 2)(x – 1) – x + 4 Sign-Charts and Inequalities I
  75. 75. Example F. Solve x – 2 2 < x – 1 3 Set the inequality to 0, x – 2 2 x – 1 3 < 0 Put the expression into factored form, x – 2 2 x – 1 3 = (x – 2)(x – 1) 2(x – 1) – 3(x – 2) = (x – 2)(x – 1) – x + 4 Hence the inequality is (x – 2)(x – 1) – x + 4 < 0 Draw the sign chart by sampling x = 0, 3/2, 3, 5 It has a root at x = 4, and it's undefined at x = 1, 2. 410 5 + + + – – + + + + 23/2 3 UDF UDF (x – 2)(x – 1) – x + 4 Sign-Charts and Inequalities I
  76. 76. Example F. Solve x – 2 2 < x – 1 3 Set the inequality to 0, x – 2 2 x – 1 3 < 0 Put the expression into factored form, x – 2 2 x – 1 3 = (x – 2)(x – 1) 2(x – 1) – 3(x – 2) = (x – 2)(x – 1) – x + 4 Hence the inequality is (x – 2)(x – 1) – x + 4 < 0 Draw the sign chart by sampling x = 0, 3/2, 3, 5 It has a root at x = 4, and it's undefined at x = 1, 2. 410 5 + + + – – + + + + – – – – 23/2 3 UDF UDF (x – 2)(x – 1) – x + 4 Sign-Charts and Inequalities I
  77. 77. Example F. Solve x – 2 2 < x – 1 3 Set the inequality to 0, x – 2 2 x – 1 3 < 0 Put the expression into factored form, x – 2 2 x – 1 3 = (x – 2)(x – 1) 2(x – 1) – 3(x – 2) = (x – 2)(x – 1) – x + 4 Hence the inequality is (x – 2)(x – 1) – x + 4 < 0 Draw the sign chart by sampling x = 0, 3/2, 3, 5 It has a root at x = 4, and it's undefined at x = 1, 2. 410 5 + + + – – + + + + – – – – 23/2 3 UDF UDF (x – 2)(x – 1) – x + 4 The answer are the shaded negative regions, i.e. (1, 2) U [4 ∞). Sign-Charts and Inequalities I
  78. 78. Inequalities

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