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Absolute Value Equations and · PDF file Absolute Value De nition 1 The absolute value of any number x is the distance between x and the zero. We denote it by jxj. Example 2 j2j= distance

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  • Absolute Value Equations and Inequalities

    Dr. Abdulla Eid

    College of Science

    TC2MA243: Topics in Algebra

    Dr. Abdulla Eid (University of Bahrain) 1 / 1

  • Learning Objectives:

    Solve equations involving absolute values.

    Solve inequality involving absolute values.

    Dr. Abdulla Eid (University of Bahrain) 2 / 1

  • Learning Objectives:

    Solve equations involving absolute values.

    Solve inequality involving absolute values.

    Dr. Abdulla Eid (University of Bahrain) 2 / 1

  • Absolute Value

    Definition 1

    The absolute value of any number x is the distance between x and the zero.

    We denote it by |x |.

    Example 2

    |2| = distance between 2 and 0 = 2.

    | − 3| = distance between -3 and 0 = 3.

    |0| = distance between 0 and 0 = 0.

    | − 2| = distance between -2 and 0 = 2.

    Note: The absolute value |x | is always non–negative, i.e., |x | ≥ 0.

    Dr. Abdulla Eid (University of Bahrain) 3 / 1

  • Absolute Value

    Definition 1

    The absolute value of any number x is the distance between x and the zero. We denote it by |x |.

    Example 2

    |2| = distance between 2 and 0 = 2.

    | − 3| = distance between -3 and 0 = 3.

    |0| = distance between 0 and 0 = 0.

    | − 2| = distance between -2 and 0 = 2.

    Note: The absolute value |x | is always non–negative, i.e., |x | ≥ 0.

    Dr. Abdulla Eid (University of Bahrain) 3 / 1

  • Absolute Value

    Definition 1

    The absolute value of any number x is the distance between x and the zero. We denote it by |x |.

    Example 2

    |2| =

    distance between 2 and 0 = 2.

    | − 3| = distance between -3 and 0 = 3.

    |0| = distance between 0 and 0 = 0.

    | − 2| = distance between -2 and 0 = 2.

    Note: The absolute value |x | is always non–negative, i.e., |x | ≥ 0.

    Dr. Abdulla Eid (University of Bahrain) 3 / 1

  • Absolute Value

    Definition 1

    The absolute value of any number x is the distance between x and the zero. We denote it by |x |.

    Example 2

    |2| = distance between 2 and 0

    = 2.

    | − 3| = distance between -3 and 0 = 3.

    |0| = distance between 0 and 0 = 0.

    | − 2| = distance between -2 and 0 = 2.

    Note: The absolute value |x | is always non–negative, i.e., |x | ≥ 0.

    Dr. Abdulla Eid (University of Bahrain) 3 / 1

  • Absolute Value

    Definition 1

    The absolute value of any number x is the distance between x and the zero. We denote it by |x |.

    Example 2

    |2| = distance between 2 and 0 = 2.

    | − 3| = distance between -3 and 0 = 3.

    |0| = distance between 0 and 0 = 0.

    | − 2| = distance between -2 and 0 = 2.

    Note: The absolute value |x | is always non–negative, i.e., |x | ≥ 0.

    Dr. Abdulla Eid (University of Bahrain) 3 / 1

  • Absolute Value

    Definition 1

    The absolute value of any number x is the distance between x and the zero. We denote it by |x |.

    Example 2

    |2| = distance between 2 and 0 = 2.

    | − 3| =

    distance between -3 and 0 = 3.

    |0| = distance between 0 and 0 = 0.

    | − 2| = distance between -2 and 0 = 2.

    Note: The absolute value |x | is always non–negative, i.e., |x | ≥ 0.

    Dr. Abdulla Eid (University of Bahrain) 3 / 1

  • Absolute Value

    Definition 1

    The absolute value of any number x is the distance between x and the zero. We denote it by |x |.

    Example 2

    |2| = distance between 2 and 0 = 2.

    | − 3| = distance between -3 and 0

    = 3.

    |0| = distance between 0 and 0 = 0.

    | − 2| = distance between -2 and 0 = 2.

    Note: The absolute value |x | is always non–negative, i.e., |x | ≥ 0.

    Dr. Abdulla Eid (University of Bahrain) 3 / 1

  • Absolute Value

    Definition 1

    The absolute value of any number x is the distance between x and the zero. We denote it by |x |.

    Example 2

    |2| = distance between 2 and 0 = 2.

    | − 3| = distance between -3 and 0 = 3.

    |0| = distance between 0 and 0 = 0.

    | − 2| = distance between -2 and 0 = 2.

    Note: The absolute value |x | is always non–negative, i.e., |x | ≥ 0.

    Dr. Abdulla Eid (University of Bahrain) 3 / 1

  • Absolute Value

    Definition 1

    The absolute value of any number x is the distance between x and the zero. We denote it by |x |.

    Example 2

    |2| = distance between 2 and 0 = 2.

    | − 3| = distance between -3 and 0 = 3.

    |0| =

    distance between 0 and 0 = 0.

    | − 2| = distance between -2 and 0 = 2.

    Note: The absolute value |x | is always non–negative, i.e., |x | ≥ 0.

    Dr. Abdulla Eid (University of Bahrain) 3 / 1

  • Absolute Value

    Definition 1

    The absolute value of any number x is the distance between x and the zero. We denote it by |x |.

    Example 2

    |2| = distance between 2 and 0 = 2.

    | − 3| = distance between -3 and 0 = 3.

    |0| = distance between 0 and 0

    = 0.

    | − 2| = distance between -2 and 0 = 2.

    Note: The absolute value |x | is always non–negative, i.e., |x | ≥ 0.

    Dr. Abdulla Eid (University of Bahrain) 3 / 1

  • Absolute Value

    Definition 1

    The absolute value of any number x is the distance between x and the zero. We denote it by |x |.

    Example 2

    |2| = distance between 2 and 0 = 2.

    | − 3| = distance between -3 and 0 = 3.

    |0| = distance between 0 and 0 = 0.

    | − 2| = distance between -2 and 0 = 2.

    Note: The absolute value |x | is always non–negative, i.e., |x | ≥ 0.

    Dr. Abdulla Eid (University of Bahrain) 3 / 1

  • Absolute Value

    Definition 1

    The absolute value of any number x is the distance between x and the zero. We denote it by |x |.

    Example 2

    |2| = distance between 2 and 0 = 2.

    | − 3| = distance between -3 and 0 = 3.

    |0| = distance between 0 and 0 = 0.

    | − 2| =

    distance between -2 and 0 = 2.

    Note: The absolute value |x | is always non–negative, i.e., |x | ≥ 0.

    Dr. Abdulla Eid (University of Bahrain) 3 / 1

  • Absolute Value

    Definition 1

    The absolute value of any number x is the distance between x and the zero. We denote it by |x |.

    Example 2

    |2| = distance between 2 and 0 = 2.

    | − 3| = distance between -3 and 0 = 3.

    |0| = distance between 0 and 0 = 0.

    | − 2| = distance between -2 and 0

    = 2.

    Note: The absolute value |x | is always non–negative, i.e., |x | ≥ 0.

    Dr. Abdulla Eid (University of Bahrain) 3 / 1

  • Absolute Value

    Definition 1

    The absolute value of any number x is the distance between x and the zero. We denote it by |x |.

    Example 2

    |2| = distance between 2 and 0 = 2.

    | − 3| = distance between -3 and 0 = 3.

    |0| = distance between 0 and 0 = 0.

    | − 2| = distance between -2 and 0 = 2.

    Note: The absolute value |x | is always non–negative, i.e., |x | ≥ 0.

    Dr. Abdulla Eid (University of Bahrain) 3 / 1

  • Absolute Value

    Definition 1

    The absolute value of any number x is the distance between x and the zero. We denote it by |x |.

    Example 2

    |2| = distance between 2 and 0 = 2.

    | − 3| = distance between -3 and 0 = 3.

    |0| = distance between 0 and 0 = 0.

    | − 2| = distance between -2 and 0 = 2.

    Note: The absolute value |x | is always non–negative, i.e., |x | ≥ 0.

    Dr. Abdulla Eid (University of Bahrain) 3 / 1

  • Properties of Absolute Values

    1 |ab| = |a| · |b|.

    2

    ∣∣ a b

    ∣∣ = |a||b| . 3 |a− b| = |b− a|.

    4 |a+ b|≤|a|+ |b|.

    5 −|a| ≤ a ≤ |a|.

    Dr. Abdulla Eid (University of Bahrain) 4 / 1

  • Properties of Absolute Values

    1 |ab| = |a| · |b|.

    2

    ∣∣ a b

    ∣∣ = |a||b| .

    3 |a− b| = |b− a|.

    4 |a+ b|≤|a|+ |b|.

    5 −|a| ≤ a ≤ |a|.

    Dr. Abdulla Eid (University of Bahrain) 4 / 1

  • Properties of Absolute Values

    1 |ab| = |a| · |b|.

    2

    ∣∣ a b

    ∣∣ = |a||b| . 3 |a− b| = |b− a|.

    4 |a+ b|≤|a|+ |b|.

    5 −|a| ≤ a ≤ |a|.

    Dr. Abdulla Eid (University of Bahrain) 4 / 1

  • Properties of Absolute Values

    1 |ab| = |a| · |b|.

    2

    ∣∣ a b

    ∣∣ = |a||b| . 3 |a− b| = |b− a|.

    4 |a+ b|≤|a|+ |b|.

    5 −|a| ≤ a ≤ |a|.

    Dr. Abdulla Eid (University of Bahrain) 4 / 1

  • Properties of Absolute Values

    1 |ab| = |a| · |b|.

    2

    ∣∣ a b

    ∣∣ = |a||b| . 3 |a− b| = |b− a|.

    4 |a+ b|≤|a|+ |b|.

    5 −|a| ≤ a ≤ |a|.

    Dr. Abdulla Eid (University of Bahrain) 4 / 1

  • Properties of Absolute Values

    1 |ab| = |a| · |b|.

    2

    ∣∣ a b

    ∣∣ = |a||b| . 3 |a− b| = |b− a|.

    4 |a+ b|≤|a|+ |b|.

    5 −|a| ≤ a ≤ |a|.

    Dr. Abdulla Eid (University of Bahrain) 4 / 1

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