k, which is equivalent to x < -k or x > k. It provides examples of solving absolute value inequalities by rewriting them as two separate inequalities and solving each one."> k, which is equivalent to x < -k or x > k. It provides examples of solving absolute value inequalities by rewriting them as two separate inequalities and solving each one.">
Nothing Special   »   [go: up one dir, main page]
Scribd Logo
Upload

Welcome to Scribd!

  • 53 views

    Absolute Value Inequalities

    Uploaded by

    Lazy Bandit
    The document discusses absolute value inequalities. There are two cases: 1) |x| < k, which is equivalent to -k < x < k, and 2) |x| > k, which is equivalent to x < -k or x > k. It provides examples of solving absolute value inequalities by rewriting them as two separate inequalities and solving each one.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    Absolute Value Inequalities

    Uploaded by

    Lazy Bandit
    53 views3 pages
    The document discusses absolute value inequalities. There are two cases: 1) |x| < k, which is equivalent to -k < x < k, and 2) |x| > k, which is equivalent to x < -k or x > k. It provides examples of solving absolute value inequalities by rewriting them as two separate inequalities and solving each one.

    Original Description:

    Class notes on absolute value inequalities.

    Original Title

    ABSOLUTE VALUE INEQUALITIES

    Copyright

    © © All Rights Reserved

    Available Formats

    PDF, TXT or read online from Scribd

    Did you find this document useful?

    Is this content inappropriate?

    The document discusses absolute value inequalities. There are two cases: 1) |x| < k, which is equivalent to -k < x < k, and 2) |x| > k, which is equivalent to x < -k or x > k. It provides examples of solving absolute value inequalities by rewriting them as two separate inequalities and solving each one.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    53 views3 pages

    Absolute Value Inequalities

    Uploaded by

    Lazy Bandit
    The document discusses absolute value inequalities. There are two cases: 1) |x| < k, which is equivalent to -k < x < k, and 2) |x| > k, which is equivalent to x < -k or x > k. It provides examples of solving absolute value inequalities by rewriting them as two separate inequalities and solving each one.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 3

    ABSOLUTE VALUE INEQUALITIES

    Reminder: The absolute value of the real number 𝑥 is the distance on the
    number line from the origin to the number 𝑥.

    There are two cases:

    a) Case 1: |𝑥| < 𝑘

    Let 𝑘 > 0, then |𝑥| < 𝑘 means the distance from the origin to the number 𝑥
    is less than 𝑘.

    
    | 
    | | 
    k 0 k

    Thus |𝑥| < 𝑘 is equivalent to −𝑘 < 𝑥 < 𝑘.

    Property 1-On absolute value inequalities

    For any real number 𝑘 > 0,

    |𝑥| < 𝑘 is equivalent to −𝑘 < 𝑥 < 𝑘

    and

    |𝑥| ≤ 𝑘 is equivalent to −𝑘 ≤ 𝑥 ≤ 𝑘

    EXAMPLE 1

    Solve:

    i. |2𝑥 + 1| < 5
    ii. |6𝑥 − 1| ≤ 3

    Solutions:

    i. |2𝑥 + 1| < 5 is equivalent to


    −5 < 2𝑥 + 1 < 5
     −5 − 1 < 2𝑥 + 1 − 1 < 5 − 1
     −6 < 2𝑥 < 4
    −6 2𝑥 4
     < <2
    2 2
     −3 < 𝑥 < 2

    1
    ∴ 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑠𝑒𝑡 = (−3, 2) in interval notation.

    ii. |6𝑥 − 1| ≤ 3 is equivalent to


     −3 ≤ 6𝑥 − 1 ≤ 3
     −3 + 1 ≤ 6𝑥 − 1 + 1 ≤ 3 + 1
     −2 ≤ 6𝑥 ≤ 4
    −2 6𝑥 4
     ≤ ≤6
    6 6
    1 2
     −3 ≤ 𝑥 ≤ 3

    1 2
    ∴ 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑠𝑒𝑡 = [− 3 , 3] in interval notation.

    b) Case 2: |𝑥| > 0


    Let 𝑘 > 0, then |𝑥| > 𝑘 means the distance from origin to the number
    𝑥 is greater than 𝑘.

    | | 


    | 
    
    k 0 k

    Hence, |𝑥| > 𝑘 is equivalent to double disjoint sets 𝑥 < −𝑘 or 𝑥 > 𝑘.

    Property 2- On absolute value inequalities

    For any real number 𝑘 > 0,

    |𝑥| > 𝑘 is equivalent to 𝑥 < −𝑘 𝑜𝑟 𝑥 > 𝑘

    and

    |𝑥| ≥ 𝑘 is equivalent to 𝑥 ≤ −𝑘 𝑜𝑟 𝑥 ≥ 𝑘

    EXAMPLE 2

    Solve |5𝑥 − 10| > 20.

    Solution:

    We write the inequality as two separate inequalities

    5𝑥 − 10 < −20 or 5𝑥 − 10 > 20

    and solve each one for 𝑥:

    2
    either 5𝑥 − 10 < −20

     5𝑥 < −20 + 10
     5𝑥 < −10
    5𝑥 −10
     <
    5 5
     𝑥 < −2

    Or 5𝑥 − 10 > 20

     5𝑥 > 20 + 10
     5𝑥 > 30
    5𝑥 30
     >
    5 5
     𝑥>6

    Hence, the solution set of the inequality |5𝑥 − 10| > 20 is:

    𝑥 < −2 𝑜𝑟 𝑥 > 6 or

    (−∞, −2) ∪ (6, ∞).

    EXAMPLE 3

    Solve the inequality:

    𝑥−2
    | | ≥ 4.
    𝑥+3

    You might also like