1.7_Linear and Absolute Value Inequalities

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    College Algebra: Section 1.7

    Equations and InequalitiesInvolving Absolute Value

    Objectives of this Section

    Solve Equations Involving Absolute Value

    Solve Inequalities Involving Absolute Value

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    Equations Involving Absolute Value

    If the absolute value of an expression equals

    some positive numbera, then the expression

    itself equals eithera or -a. Thus,u a u a u a is equivalent to or

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    Solve: 2 3 11x

    2 3 11x

    2 3 11 2 3 11x x or

    2 14x x 7

    2 8x x 4

    Solution set: {-4, 7}

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    u a a u a

    u a a u a

    is equivalent to

    is equivalent to

    Theorem

    In other words, |u| < a is equivalent

    to -a < u and u < a.

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    Ex. Solve y + 21 7

    - 21 -21y -14

    (-14) + 21 7

    7 7

    1. Draw the river

    2. Subtract 21 fromboth sides

    3. Simplify4. Check your answer

    5. Graph the solution

    -14 -13-15

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    Ex. Solve 8y + 3 > 9y - 14

    o17 1816

    - 8y - 8y3 > y - 14

    + 14 + 1417 > y

    y

    < 178(16) + 3 > 9(16)14131 > 130

    1. Draw the river2. Subtract 8y from both

    sides

    3. Simplify

    4. Add 14 to both sides5. Simplify

    6. Rewrite inequality

    with the variable first

    7. Check your answer8. Graph the solution

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    Ex. Solve 8y + 3 > 9y - 14

    o17 1816

    - 8y - 8y3 > y - 14

    + 14 + 1417 > y

    y

    < 178(16) + 3 = 9(16)14131 > 130

    Big Tip!!!

    At the end of solving

    inequality, always put the

    variable at the LEFT hand

    side. Then arrow of the

    inequality sign tells you the

    correct graph

    y < 17

    The graph should toward to

    the left.

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    To Solve the Absolute Value Inequalities

    1. Isolate the absolute value expression.

    2. Make sure the absolute value inequality can be defined.

    3. For any defined absolute value inequality with template:|X| a, where a> 0 and = , ,

    write the corresponding compound inequalities by following the

    rules:

    a) if |X| > a or |X| a, meaning greator, leaving

    the jail

    b) set jail boundaries as a and ac) write compound inequalities as

    X a meaning stay left to the left

    boundary or right to the right boundary

    d) if |X| < a or |X| a, meaning less thand, going

    to the jail

    e) set jail boundaries as a and a

    f) write compound inequalities as

    a< X < a (and) meaning stay between the two

    boundaries

    4. Solve the converted compound inequalities.

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    Solve | x3 | < 5

    Absolute Value Inequality

    (1) The absolute value expression is isolated(2) It is a well defined absolute value inequality

    (3) It is a less thand inequality. (go to the jail)

    a) set jail boundaries: 5, and 5

    b) write compound inequalities: (stay in between the jailboundaries)

    5 < x3 < 5

    2 < x < 8

    +3 +3 +3

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    You try this!

    Solve | x + 4 | < 1

    Absolute Value Inequality

    (1) The absolute value expression is isolated

    (2) It is a well defined absolute value inequality

    (3) It is a less thand inequality. (go to the jail)

    a) set jail boundaries: 1, and 1

    b) write compound inequalities: (stay in between the jail

    boundaries)

    1 < x + 4 < 1

    5 < x

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    Solve | 2x + 3 |3 2

    Absolute Value Inequality

    (1) The absolute value expression is NOT isolated

    | 2x + 3 |3 2

    | 2x+ 3 | 5

    (2) It is a well defined absolute value inequality(3) It is a greator inequality. (leaving the jail)

    a) set jail boundaries: 5, and 5

    b) write compound inequalities: (stay left to left boundary

    and right to the right boundary)

    2x + 3 5 or 2x + 3 5

    2x 8 or 2x 2

    x 4 or x 1

    3 3

    + 3 + 3

    3 3

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    You try this!

    Solve | 4x3 | + 5 8

    Absolute Value Inequality

    (1) The absolute value expression is NOT isolated

    | 4x3| + 5 8

    | 4x3 3

    (2) It is a well defined absolute value inequality(3) It is a greator inequality. (leaving the jail)

    a) set jail boundaries: 3, and 3

    b) write compound inequalities: (stay left to left boundary

    and right to the right boundary)

    4x3 3 or 4x3 3

    4x 0 or 4x 6

    x 0 or x 3/2

    + 3 + 3

    5 5

    + 3 + 3

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    Summary

    1. Before solving absolute value equations or inequalities, you

    MUST isolate the absolute value expression and check themare defined or not.

    2. Remember telling yourself the jail story. It will help you set

    up the correct equations or compound inequalities.

    3. Solve the equations or compound inequalities and dont

    forget the learned knowledge such as when multiplying or

    dividing a negative number, you need flip the inequality

    sign.

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    Solve: 3 1 5x

    3 1 5x

    5 3 1 5x

    4 3 6x

    4

    32x

    Solution Set:

    2,3

    42

    3

    4| orxx

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    u a u a u a

    u a u a u a

    is equivalent to or

    is equivalent to or

    Ifa is any positive number, then

    Theorem

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    Solve: 4 3 15x

    4 3 15x

    4 3 15x or 4 3 15x

    4 18x

    x 9

    2

    4 12x

    x 3

    ),3[2

    9,3

    2

    9|

    orxorxx