Inequality with Fractions

If x is a number such that and , then which of the following is true?

  1. or
  2. or


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We have . We need to consider two cases, x > 0 and x < 0, because when we multiply both sides of an inequality by a negative number (and we feel in our bones that we�re going to do this), the sense of the inequality is reversed.

If x < 0, then is always true, so 1 > 2x and . From we get:
-3x > 1 so the stronger condition holds.

If x > 0, then is always true and -3x < 1 so . From , we get:
1 < 2x so the stronger condition holds.

So the two cases lead to the two possibilities listed in (e).

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