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Math Help - inequality in intervals

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    inequality in intervals



    where the "?"s are, is either "(" or "[".
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  2. #2
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    \frac{1}{x} < 4.

    First, it should be abundantly obvious that x \neq 0.

    To solve this inequality for x, you will need to consider two cases, the first being when x is positive, and the second when x is negative, since multiplying/dividing by a negative reverses the inequality sign.


    Case 1: x > 0.

    \frac{1}{x} < 4

    1 < 4x

    \frac{1}{4} < x.

    Since x > 0 and x > \frac{1}{4}, putting it together gives x > \frac{1}{4}, or x \in \left(\frac{1}{4}, \infty\right).


    Case 2: x < 0

    \frac{1}{x} < 4

    1 > 4x

    \frac{1}{4} > x.

    So x < 0 and x < \frac{1}{4}. Putting it together gives x < 0, or x \in (-\infty, 0).


    Therefore, our final solution is

    x \in (-\infty, 0) \cup \left(\frac{1}{4}, \infty\right). You can check this by graphing the functions y = \frac{1}{x} and y = 4 and making sure that everything below the line y = 4 are the x values listed above.
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