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Math Help - Absolute Value

  1. #1
    Member srirahulan's Avatar
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    Lightbulb Absolute Value

    \mid 2x+5 \mid - \mid x+6 \mid < 2 Find out the suitable value range for x?
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  2. #2
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    Re: Absolute Value

    Quote Originally Posted by srirahulan View Post
    \mid 2x+5 \mid - \mid x+6 \mid < 2 Find out the suitable value range for x?
    1. Use the definition of the absolute value:

    |2x+5| = \left \lbrace \begin{array}{ccc}2x+5, & 2x+5 \ge 0 \implies & x\ge -\frac52 \\ -(2x+5), & 2x+5 < 0 \implies & x < -\frac52 \end{array} \right.

    |x+6| = \left \lbrace \begin{array}{ccc}x+6, & x+6 \ge 0 \implies & x\ge -6 \\ -(x+6), & x+6 < 0 \implies & x < -6 \end{array} \right.

    2. You'll get 3 intervals with different inequalities:

    (-\infty, -6)
    \left[-6, -\frac52 \right)
    \left[ -\frac52, +\infty \right)

    3. Solve the corresponding 3 inequalities, check the result against the domain.

    4. You should come out with x \in \left(- \frac{13}3 , 3 \right)

    5. Merry Christmas!
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  3. #3
    Member srirahulan's Avatar
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    Re: Absolute Value

    Thank You and Merry Christmas.
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