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1.4 sign charts and inequalities i
 

1.4 sign charts and inequalities i

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    1.4 sign charts and inequalities i 1.4 sign charts and inequalities i Presentation Transcript

    • Sign-Charts and Inequalities I
    • Sign-Charts and Inequalities I
      We use the factored polynomials or rational expressions to determine the signs of the outputs.
    • Sign-Charts and Inequalities I
      We use the factored polynomials or rational expressions to determine the signs of the outputs.
      That is, given a formula f and a value for x, factor f to determine if the output is positive (f > 0) or negative (f < 0) when f is evaluated with x.
    • Sign-Charts and Inequalities I
      We use the factored polynomials or rational expressions to determine the signs of the outputs.
      That is, given a formula f and a value for x, factor f to determine if the output is positive (f > 0) or negative (f < 0) when f is evaluated with x.
      Example A: Determine the outcome is + or – for
      x2 – 2x – 3 if x = -3/2, -1/2.
    • Sign-Charts and Inequalities I
      We use the factored polynomials or rational expressions to determine the signs of the outputs.
      That is, given a formula f and a value for x, factor f to determine if the output is positive (f > 0) or negative (f < 0) when f is evaluated with x.
      Example A: Determine the outcome is + or – for
      x2 – 2x – 3 if x = -3/2, -1/2.
      In factored form x2 – 2x – 3 = (x – 3)(x + 1)
    • Sign-Charts and Inequalities I
      We use the factored polynomials or rational expressions to determine the signs of the outputs.
      That is, given a formula f and a value for x, factor f to determine if the output is positive (f > 0) or negative (f < 0) when f is evaluated with x.
      Example A: Determine the outcome is + or – for
      x2 – 2x – 3 if x = -3/2, -1/2.
      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
      We use the factored polynomials or rational expressions to determine the signs of the outputs.
      That is, given a formula f and a value for x, factor f to determine if the output is positive (f > 0) or negative (f < 0) when f is evaluated with x.
      Example A: Determine the outcome is + or – for
      x2 – 2x – 3 if x = -3/2, -1/2.
      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
      We use the factored polynomials or rational expressions to determine the signs of the outputs.
      That is, given a formula f and a value for x, factor f to determine if the output is positive (f > 0) or negative (f < 0) when f is evaluated with x.
      Example A: Determine the outcome is + or – for
      x2 – 2x – 3 if x = -3/2, -1/2.
      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
      We use the factored polynomials or rational expressions to determine the signs of the outputs.
      That is, given a formula f and a value for x, factor f to determine if the output is positive (f > 0) or negative (f < 0) when f is evaluated with x.
      Example A: Determine the outcome is + or – for
      x2 – 2x – 3 if x = -3/2, -1/2.
      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
      Example B: Determine the outcome is + or – for
      if x = -3/2, -1/2.
      x2 – 2x – 3
      x2 + x – 2
    • Sign-Charts and Inequalities I
      Example B: Determine the outcome is + or – for
      if x = -3/2, -1/2.
      x2 – 2x – 3
      x2 + x – 2
      (x – 3)(x + 1)
      x2 – 2x – 3
      =
      In factored form
      x2 + x – 2
      (x – 1)(x + 2)
    • Sign-Charts and Inequalities I
      Example B: Determine the outcome is + or – for
      if x = -3/2, -1/2.
      x2 – 2x – 3
      x2 + x – 2
      (x – 3)(x + 1)
      x2 – 2x – 3
      =
      In factored form
      x2 + x – 2
      (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 the outcome is + or – for
      if x = -3/2, -1/2.
      x2 – 2x – 3
      x2 + x – 2
      (x – 3)(x + 1)
      x2 – 2x – 3
      =
      In factored form
      x2 + x – 2
      (x – 1)(x + 2)
      Hence, for x = -3/2:
      (x – 3)(x + 1)
      (–)(–)
      < 0
      =
      (–)(+)
      (x – 1)(x + 2)
    • Sign-Charts and Inequalities I
      Example B: Determine the outcome is + or – for
      if x = -3/2, -1/2.
      x2 – 2x – 3
      x2 + x – 2
      (x – 3)(x + 1)
      x2 – 2x – 3
      =
      In factored form
      x2 + x – 2
      (x – 1)(x + 2)
      Hence, for x = -3/2:
      (x – 3)(x + 1)
      (–)(–)
      < 0
      =
      (–)(+)
      (x – 1)(x + 2)
      For x = -1/2:
      (x – 3)(x + 1)
      (–)(+)
      =
      (–)(+)
      (x – 1)(x + 2)
    • Sign-Charts and Inequalities I
      Example B: Determine the outcome is + or – for
      if x = -3/2, -1/2.
      x2 – 2x – 3
      x2 + x – 2
      (x – 3)(x + 1)
      x2 – 2x – 3
      =
      In factored form
      x2 + x – 2
      (x – 1)(x + 2)
      Hence, for x = -3/2:
      (x – 3)(x + 1)
      (–)(–)
      < 0
      =
      (–)(+)
      (x – 1)(x + 2)
      For x = -1/2:
      (x – 3)(x + 1)
      (–)(+)
      > 0
      =
      (–)(+)
      (x – 1)(x + 2)
    • Sign-Charts and Inequalities I
      Example B: Determine the outcome is + or – for
      if x = -3/2, -1/2.
      x2 – 2x – 3
      x2 + x – 2
      (x – 3)(x + 1)
      x2 – 2x – 3
      =
      In factored form
      x2 + x – 2
      (x – 1)(x + 2)
      Hence, for x = -3/2:
      (x – 3)(x + 1)
      (–)(–)
      < 0
      =
      (–)(+)
      (x – 1)(x + 2)
      For x = -1/2:
      (x – 3)(x + 1)
      (–)(+)
      > 0
      =
      (–)(+)
      (x – 1)(x + 2)
      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
      Here is an example, the sign chart of f = x – 1:




      +
      +
      f = 0
      x – 1
      1
    • Sign-Charts and Inequalities I
      Here is an example, the sign chart of f = x – 1:




      +
      +
      f = 0
      x – 1
      1
      The "+" indicates the region where the output is positive i.e. if 1 < x.
    • Sign-Charts and Inequalities I
      Here is an example, the sign chart of f = x – 1:




      +
      +
      f = 0
      x – 1
      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
      Here is an example, the sign chart of f = x – 1:




      +
      +
      f = 0
      x – 1
      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.
      Construction of the sign-chart of f.
    • Sign-Charts and Inequalities I
      Here is an example, the sign chart of f = x – 1:




      +
      +
      f = 0
      x – 1
      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.
      Construction of the sign-chart of f.
      I. Solve for f = 0 (and denominator = 0) if there is any denominator.
    • Sign-Charts and Inequalities I
      Here is an example, the sign chart of f = x – 1:




      +
      +
      f = 0
      x – 1
      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.
      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.
    • Sign-Charts and Inequalities I
      Here is an example, the sign chart of f = x – 1:




      +
      +
      f = 0
      x – 1
      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.
      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.
    • Sign-Charts and Inequalities I
      Here is an example, the sign chart of f = x – 1:




      +
      +
      f = 0
      x – 1
      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.
      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.
      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
      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
      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
      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
      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:
    • Sign-Charts and Inequalities I
      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
      -2
      Select points to sample in each segment:
      Test x = - 2,
    • Sign-Charts and Inequalities I
      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
      -2
      Select points to sample in each segment:
      Test x = - 2,
      get – * – = + .
      Hence the segment
      is positive. Draw +
      sign over it.
    • Sign-Charts and Inequalities I
      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
      -2
      Select points to sample in each segment:
      Test x = - 2,
      get – * – = + .
      Hence the segment
      is positive. Draw +
      sign over it.
    • Sign-Charts and Inequalities I
      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
      -2
      0
      Select points to sample in each segment:
      Test x = 0,
      get – * + = –.
      Hence this segment
      is negative.
      Put – over it.
      Test x = - 2,
      get – * – = + .
      Hence the segment
      is positive. Draw +
      sign over it.
    • Sign-Charts and Inequalities I
      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
      -2
      0
      Select points to sample in each segment:
      Test x = 0,
      get – * + = –.
      Hence this segment
      is negative.
      Put – over it.
      Test x = - 2,
      get – * – = + .
      Hence the segment
      is positive. Draw +
      sign over it.
    • Sign-Charts and Inequalities I
      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
      -2
      5
      0
      Select points to sample in each segment:
      Test x = 5,
      get + * + = +.
      Hence this segment
      is positive.
      Put + over it.
      Test x = 0,
      get – * + = –.
      Hence this segment
      is negative.
      Put – over it.
      Test x = - 2,
      get – * – = + .
      Hence the segment
      is positive. Draw +
      sign over it.
    • Sign-Charts and Inequalities I
      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
      -2
      5
      0
      Select points to sample in each segment:
      Test x = 5,
      get + * + = +.
      Hence this segment
      is positive.
      Put + over it.
      Test x = 0,
      get – * + = –.
      Hence this segment
      is negative.
      Put – over it.
      Test x = - 2,
      get – * – = + .
      Hence the segment
      is positive. Draw +
      sign over it.
    • Sign-Charts and Inequalities I
      (x – 3)
      Example D: Make the sign chart of f =
      (x – 1)(x + 2)
    • Sign-Charts and Inequalities I
      (x – 3)
      Example D: Make the sign chart of f =
      (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
      (x – 3)
      Example D: Make the sign chart of f =
      (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
      (x – 3)
      Example D: Make the sign chart of f =
      (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
      (x – 3)
      Example D: Make the sign chart of f =
      (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)
      UDF
      UDF
      f=0
      (x – 1)(x + 2)
      -2
      1
      3
    • Sign-Charts and Inequalities I
      (x – 3)
      Example D: Make the sign chart of f =
      (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.
      Select a point to sample in each segment:
      (x – 3)
      UDF
      UDF
      f=0
      (x – 1)(x + 2)
      -2
      1
      3
      -3
      0
      2
      4
    • Sign-Charts and Inequalities I
      (x – 3)
      Example D: Make the sign chart of f =
      (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.
      Select a point to sample in each segment:
      Test x = -3,
      we've a
      ( – )
      = –
      ( – )( – )
      segment.
      (x – 3)
      UDF
      UDF
      f=0
      (x – 1)(x + 2)
      -2
      1
      3
      -3
      0
      2
      4
    • Sign-Charts and Inequalities I
      (x – 3)
      Example D: Make the sign chart of f =
      (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.
      Select a point to sample in each segment:
      Test x = -3,
      we've a
      Test x = 0,
      we've a
      ( – )
      ( – )
      = –
      = +
      ( – )( – )
      ( – )( + )
      segment.
      segment.
      (x – 3)
      UDF
      UDF
      f=0
      (x – 1)(x + 2)
      -2
      1
      3
      -3
      0
      2
      4
    • Sign-Charts and Inequalities I
      (x – 3)
      Example D: Make the sign chart of f =
      (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.
      Select a point to sample in each segment:
      Test x = -3,
      we've a
      Test x = 0,
      we've a
      Test x = 2,
      we've a
      ( – )
      ( – )
      ( – )
      = –
      = +
      = –
      ( – )( – )
      ( – )( + )
      ( + )( + )
      segment.
      segment.
      segment.
      (x – 3)
      UDF
      UDF
      f=0
      (x – 1)(x + 2)
      -2
      1
      3
      -3
      0
      2
      4
    • Sign-Charts and Inequalities I
      (x – 3)
      Example D: Make the sign chart of f =
      (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.
      Select a point to sample in each segment:
      Test x = -3,
      we've a
      Test x = 0,
      we've a
      Test x = 2,
      we've a
      Test x = 4,
      we've a
      ( – )
      ( – )
      ( – )
      ( + )
      = –
      = +
      = –
      = +
      ( – )( – )
      ( – )( + )
      ( + )( + )
      ( + )( + )
      segment.
      segment.
      segment.
      segment.
      – – –– + + + – –– + + + +
      (x – 3)
      UDF
      UDF
      f=0
      (x – 1)(x + 2)
      -2
      1
      3
      -3
      0
      2
      4
    • Sign-Charts and Inequalities I
      The easiest way to solve a polynomial or rational inequality is use the sign-chart.
    • Sign-Charts and Inequalities I
      The easiest way to solve a polynomial or rational inequality is use the sign-chart. To do this,
      I. set one side of the inequality to 0,
    • Sign-Charts and Inequalities I
      The easiest way to solve a polynomial or rational inequality is 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,
    • Sign-Charts and Inequalities I
      The easiest way to solve a polynomial or rational inequality is 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.
    • Sign-Charts and Inequalities I
      The easiest way to solve a polynomial or rational inequality is 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.
      Example E: Solve x2 – 3x> 4
    • Sign-Charts and Inequalities I
      The easiest way to solve a polynomial or rational inequality is 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.
      Example E: Solve x2 – 3x> 4
      Set one side to 0, we get x2 – 3x – 4> 0;
    • Sign-Charts and Inequalities I
      The easiest way to solve a polynomial or rational inequality is 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.
      Example E: Solve x2 – 3x> 4
      Set one side to 0, we get x2 – 3x – 4> 0; factor, we've
      (x – 4)(x + 1) > 0.
    • Sign-Charts and Inequalities I
      The easiest way to solve a polynomial or rational inequality is 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.
      Example E: Solve x2 – 3x> 4
      Set one side to 0, we get x2 – 3x – 4> 0; factor, we've
      (x – 4)(x + 1) > 0.
      The roots are x = -1, 4.
    • Sign-Charts and Inequalities I
      The easiest way to solve a polynomial or rational inequality is 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.
      Example E: Solve x2 – 3x> 4
      Set one side to 0, we get x2 – 3x – 4> 0; factor, we've
      (x – 4)(x + 1) > 0.
      The roots are x = -1, 4.
      Draw the sign-chart,
      (x – 4)(x + 1)
      4
      -1
    • Sign-Charts and Inequalities I
      The easiest way to solve a polynomial or rational inequality is 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.
      Example E: Solve x2 – 3x> 4
      Set one side to 0, we get x2 – 3x – 4> 0; factor, we've
      (x – 4)(x + 1) > 0.
      The roots are x = -1, 4.
      Draw the sign-chart, sample the points x = -2, 0, 5
      (x – 4)(x + 1)
      4
      -1
      -2
      5
      0
    • Sign-Charts and Inequalities I
      The easiest way to solve a polynomial or rational inequality is 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.
      Example E: Solve x2 – 3x> 4
      Set one side to 0, we get x2 – 3x – 4> 0; factor, we've
      (x – 4)(x + 1) > 0.
      The roots are x = -1, 4.
      Draw the sign-chart, sample the points x = -2, 0, 5
      + + +
      (x – 4)(x + 1)
      4
      -1
      -2
      5
      0
    • Sign-Charts and Inequalities I
      The easiest way to solve a polynomial or rational inequality is 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.
      Example E: Solve x2 – 3x> 4
      Set one side to 0, we get x2 – 3x – 4> 0; factor, we've
      (x – 4)(x + 1) > 0.
      The roots are x = -1, 4.
      Draw the sign-chart, sample the points x = -2, 0, 5
      + + + – – – –– –
      (x – 4)(x + 1)
      4
      -1
      -2
      5
      0
    • Sign-Charts and Inequalities I
      The easiest way to solve a polynomial or rational inequality is 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.
      Example E: Solve x2 – 3x> 4
      Set one side to 0, we get x2 – 3x – 4> 0; factor, we've
      (x – 4)(x + 1) > 0.
      The roots are x = -1, 4.
      Draw the sign-chart, sample the points x = -2, 0, 5
      + + + – – – –– – + + + +
      (x – 4)(x + 1)
      4
      -1
      -2
      5
      0
    • Sign-Charts and Inequalities I
      The easiest way to solve a polynomial or rational inequality is 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.
      Example E: Solve x2 – 3x> 4
      Set one side to 0, we get x2 – 3x – 4> 0; factor, we've
      (x – 4)(x + 1) > 0.
      The roots are x = -1, 4.
      Draw the sign-chart, sample the points x = -2, 0, 5
      + + + – – – –– – + + + +
      (x – 4)(x + 1)
      4
      -1
      -2
      5
      0
      The solutions the positive region, {x < -1} U {4 < x}
      4
      -1
    • Sign-Charts and Inequalities I
      The easiest way to solve a polynomial or rational inequality is 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.
      Example E: Solve x2 – 3x> 4
      Set one side to 0, we get x2 – 3x – 4> 0; factor, we've
      (x – 4)(x + 1) > 0.
      The roots are x = -1, 4.
      Draw the sign-chart, sample the points x = -2, 0, 5
      + + + – – – –– – + + + +
      (x – 4)(x + 1)
      4
      -1
      -2
      5
      0
      The solutions the positive region, {x < -1} U {4 < x}
      4
      -1
      Note: The empty dot means those numbers are excluded.
    • Sign-Charts and Inequalities I
      2
      3
      Example F: Solve
      <
      x – 2
      x – 1
    • Sign-Charts and Inequalities I
      2
      3
      Example F: Solve
      <
      x – 2
      x – 1
      2
      3
      Set the inequality to 0,
      <
      0
      x – 2
      x – 1
    • Sign-Charts and Inequalities I
      2
      3
      Example F: Solve
      <
      x – 2
      x – 1
      2
      3
      Set the inequality to 0,
      <
      0
      x – 2
      x – 1
      Put the expression into factored form,
    • Sign-Charts and Inequalities I
      2
      3
      Example F: Solve
      <
      x – 2
      x – 1
      2
      3
      Set the inequality to 0,
      <
      0
      x – 2
      x – 1
      Put the expression into factored form,
      2
      3
      2(x – 1) – 3(x – 2)
      =
      x – 2
      x – 1
      (x – 2)(x – 1)
    • Sign-Charts and Inequalities I
      2
      3
      Example F: Solve
      <
      x – 2
      x – 1
      2
      3
      Set the inequality to 0,
      <
      0
      x – 2
      x – 1
      Put the expression into factored form,
      2
      3
      2(x – 1) – 3(x – 2)
      – x + 4
      =
      =
      x – 2
      x – 1
      (x – 2)(x – 1)
      (x – 2)(x – 1)
    • Sign-Charts and Inequalities I
      2
      3
      Example F: Solve
      <
      x – 2
      x – 1
      2
      3
      Set the inequality to 0,
      <
      0
      x – 2
      x – 1
      Put the expression into factored form,
      2
      3
      2(x – 1) – 3(x – 2)
      – x + 4
      =
      =
      x – 2
      x – 1
      (x – 2)(x – 1)
      (x – 2)(x – 1)
      – x + 4
      <
      0
      Hence the inequality is
      (x – 2)(x – 1)
    • Sign-Charts and Inequalities I
      2
      3
      Example F: Solve
      <
      x – 2
      x – 1
      2
      3
      Set the inequality to 0,
      <
      0
      x – 2
      x – 1
      Put the expression into factored form,
      2
      3
      2(x – 1) – 3(x – 2)
      – x + 4
      =
      =
      x – 2
      x – 1
      (x – 2)(x – 1)
      (x – 2)(x – 1)
      – x + 4
      <
      0
      Hence the inequality is
      (x – 2)(x – 1)
      It has root at x = 4, and its undefined at x = 1, 2.
    • Sign-Charts and Inequalities I
      2
      3
      Example F: Solve
      <
      x – 2
      x – 1
      2
      3
      Set the inequality to 0,
      <
      0
      x – 2
      x – 1
      Put the expression into factored form,
      2
      3
      2(x – 1) – 3(x – 2)
      – x + 4
      =
      =
      x – 2
      x – 1
      (x – 2)(x – 1)
      (x – 2)(x – 1)
      – x + 4
      <
      0
      Hence the inequality is
      (x – 2)(x – 1)
      It has root at x = 4, and its undefined at x = 1, 2.
      Draw the sign chart by sampling x = 0, 3/2, 3, 5
    • Sign-Charts and Inequalities I
      2
      3
      Example F: Solve
      <
      x – 2
      x – 1
      2
      3
      Set the inequality to 0,
      <
      0
      x – 2
      x – 1
      Put the expression into factored form,
      2
      3
      2(x – 1) – 3(x – 2)
      – x + 4
      =
      =
      x – 2
      x – 1
      (x – 2)(x – 1)
      (x – 2)(x – 1)
      – x + 4
      <
      0
      Hence the inequality is
      (x – 2)(x – 1)
      It has root at x = 4, and its undefined at x = 1, 2.
      Draw the sign chart by sampling x = 0, 3/2, 3, 5
      – x + 4
      UDF
      UDF
      (x – 2)(x – 1)
      2
      4
      1
    • Sign-Charts and Inequalities I
      2
      3
      Example F: Solve
      <
      x – 2
      x – 1
      2
      3
      Set the inequality to 0,
      <
      0
      x – 2
      x – 1
      Put the expression into factored form,
      2
      3
      2(x – 1) – 3(x – 2)
      – x + 4
      =
      =
      x – 2
      x – 1
      (x – 2)(x – 1)
      (x – 2)(x – 1)
      – x + 4
      <
      0
      Hence the inequality is
      (x – 2)(x – 1)
      It has root at x = 4, and its undefined at x = 1, 2.
      Draw the sign chart by sampling x = 0, 3/2, 3, 5
      – x + 4
      UDF
      UDF
      (x – 2)(x – 1)
      2
      4
      1
      0
      5
      3/2
      3
    • Sign-Charts and Inequalities I
      2
      3
      Example F: Solve
      <
      x – 2
      x – 1
      2
      3
      Set the inequality to 0,
      <
      0
      x – 2
      x – 1
      Put the expression into factored form,
      2
      3
      2(x – 1) – 3(x – 2)
      – x + 4
      =
      =
      x – 2
      x – 1
      (x – 2)(x – 1)
      (x – 2)(x – 1)
      – x + 4
      <
      0
      Hence the inequality is
      (x – 2)(x – 1)
      It has root at x = 4, and its undefined at x = 1, 2.
      Draw the sign chart by sampling x = 0, 3/2, 3, 5
      + + +
      – x + 4
      UDF
      UDF
      (x – 2)(x – 1)
      2
      4
      1
      0
      5
      3/2
      3
    • Sign-Charts and Inequalities I
      2
      3
      Example F: Solve
      <
      x – 2
      x – 1
      2
      3
      Set the inequality to 0,
      <
      0
      x – 2
      x – 1
      Put the expression into factored form,
      2
      3
      2(x – 1) – 3(x – 2)
      – x + 4
      =
      =
      x – 2
      x – 1
      (x – 2)(x – 1)
      (x – 2)(x – 1)
      – x + 4
      <
      0
      Hence the inequality is
      (x – 2)(x – 1)
      It has root at x = 4, and its undefined at x = 1, 2.
      Draw the sign chart by sampling x = 0, 3/2, 3, 5
      + + + – –
      – x + 4
      UDF
      UDF
      (x – 2)(x – 1)
      2
      4
      1
      0
      5
      3/2
      3
    • Sign-Charts and Inequalities I
      2
      3
      Example F: Solve
      <
      x – 2
      x – 1
      2
      3
      Set the inequality to 0,
      <
      0
      x – 2
      x – 1
      Put the expression into factored form,
      2
      3
      2(x – 1) – 3(x – 2)
      – x + 4
      =
      =
      x – 2
      x – 1
      (x – 2)(x – 1)
      (x – 2)(x – 1)
      – x + 4
      <
      0
      Hence the inequality is
      (x – 2)(x – 1)
      It has root at x = 4, and its undefined at x = 1, 2.
      Draw the sign chart by sampling x = 0, 3/2, 3, 5
      + + + – – + + + +
      – x + 4
      UDF
      UDF
      (x – 2)(x – 1)
      2
      4
      1
      0
      5
      3/2
      3
    • Sign-Charts and Inequalities I
      2
      3
      Example F: Solve
      <
      x – 2
      x – 1
      2
      3
      Set the inequality to 0,
      <
      0
      x – 2
      x – 1
      Put the expression into factored form,
      2
      3
      2(x – 1) – 3(x – 2)
      – x + 4
      =
      =
      x – 2
      x – 1
      (x – 2)(x – 1)
      (x – 2)(x – 1)
      – x + 4
      <
      0
      Hence the inequality is
      (x – 2)(x – 1)
      It has root at x = 4, and its undefined at x = 1, 2.
      Draw the sign chart by sampling x = 0, 3/2, 3, 5
      + + + – – + + + + – –– –
      – x + 4
      UDF
      UDF
      (x – 2)(x – 1)
      2
      4
      1
      0
      5
      3/2
      3
    • Sign-Charts and Inequalities I
      2
      3
      Example F: Solve
      <
      x – 2
      x – 1
      2
      3
      Set the inequality to 0,
      <
      0
      x – 2
      x – 1
      Put the expression into factored form,
      2
      3
      2(x – 1) – 3(x – 2)
      – x + 4
      =
      =
      x – 2
      x – 1
      (x – 2)(x – 1)
      (x – 2)(x – 1)
      – x + 4
      <
      0
      Hence the inequality is
      (x – 2)(x – 1)
      It has root at x = 4, and its undefined at x = 1, 2.
      Draw the sign chart by sampling x = 0, 3/2, 3, 5
      + + + – – + + + + – –– –
      – x + 4
      UDF
      UDF
      (x – 2)(x – 1)
      2
      4
      1
      0
      5
      3/2
      3
      We want the shaded negative region,
      i.e. {1 < x < 2} U {4 < x}.