Inequality with Fractions

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.