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d Section 1.4 Absolute Value - Abdulla Eid€¦ · Dr. Abdulla Eid (University of Bahrain) Absolute...

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  • Dr. AbdullaEidSection 1.4Absolute Value

    Dr. Abdulla Eid

    College of Science

    MATHS 103: Mathematics for Business I

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  • Dr. AbdullaEid

    Absolute Value

    Definition

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

    Example

    |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.

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  • Dr. AbdullaEid

    Properties of Absolute Values

    1 |ab| = |a| · |b|.

    2

    ∣∣ ab

    ∣∣ = |a||b| .3 |a− b| = |b− a|.

    4 |a+ b|≤|a|+ |b|.

    5 −|a| ≤ a ≤ |a|.

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    Rules

    For equations (or inequalities) that involve absolute value we need to getrid of the absolute value which can be done only using the following threerules:

    1 Rule 1: |X | = a→ X = a or X = −a.

    2 Rule 2: |X | < a→ −a < X < a.

    3 Rule 3: |X | > a→ X > a or X < −a.

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    Example

    Solve |x − 3| = 2

    Solution: We solve the absolute value using rule 1 to get rid of theabsolute value.

    |x − 3| = 2x − 3 = 2 or x − 3 = −2

    x = 5 or x = 1

    Solution Set = {5, 1}.

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    Exercise

    Solve |7− 3x | = 5

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    Example

    Solve |x − 4| = −3

    Solution: Caution: The absolute value can never be negative, so in thisexample, we have to stop and we say there are no solution!Solution Set = {} = ∅.

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    Exercise

    (Old Exam Question) Solve |7x + 2| = 16.

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    Example

    Solve |x − 2| < 4

    Solution:

    |x − 2| < 4− 4 < x − 2 < 4

    − 4+ 2 < x < 4+ 2− 2 < x < 6

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    The Solution

    1- Set notationSolution Set = {x | − 2 < x < 6}

    2- Number Line notation

    3- Interval notation(−2, 6)

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  • Dr. AbdullaEid

    Exercise

    Solve |3− 2x | ≤ 5

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    Example

    Solve |x + 5| ≤ −2

    Solution: Caution: The absolute value can never be negative or less than anegative, so in this example, we have to stop and we say there are nosolution!Solution Set = {} = ∅.

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  • Dr. AbdullaEid

    Exercise

    (Old Final Exam Question) Solve |5− 6x | ≤ 1

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    Exercise

    (Old Final Exam Question) Solve |2x − 7| ≤ 9

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    Example

    Solve |x + 5| ≥ 7

    Solution:

    |x + 5| ≥ 7x + 5 ≥ 7 or x + 5 ≤ −7

    x ≥ 2 or x ≤ −12

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    The Solution

    1- Set notation

    Solution Set = {x | x ≥ 2 or x ≥ −12}

    2- Number Line notation

    3- Interval notation(−∞,−12]∪[2,∞)

    where ∪ means union of two intervals.

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    Exercise

    Solve |3x − 4| > 1

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    Example

    Solve | 3x−82 | ≥ 4

    Solution:

    |3x − 82| ≥ 4

    3x − 82≥ 4 or 3x − 8

    2≤ −4

    3x − 8 ≥ 8 or 3x − 8 ≤ −83x ≥ 16 or 3x ≤ 0

    x ≥ 163

    or x ≤ 0

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    The Solution

    1- Set notation

    Solution Set = {x | x ≥ 163

    or x ≥ 0}

    2- Number Line notation

    3- Interval notation

    (−∞, 0] ∪ [163,∞)

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    Exercise

    (Old Exam Question) Solve |x + 8|+ 3 < 2

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    Exercise

    (Old Exam Question) Solve |10x − 9| ≥ 11.

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