A rational inequality can be written as a quotient of linear factors on one side of an inequality symbol and with zero on the other side; for example:
As above for polynomials, this can be solved by use of a sign diagram. The value of the rational expression changes sign only at the critical points. A rational expression will have one more region than the number of distinct critical points. By using a map that shows where each factor is negative or positive and using the rules for dividing signed quantities, we can find the sign of the value of the rational expression in each region.x - 2 ——————— ≥ 0 x + 1
Recall that division by zero is undefined. For critical points determined by factors in the denominator, the rational expression is undefined. This means that a closed interval containing such a point would be undefined.
Example, solve:
x - 2
——————— ≥ 0,
x + 1
The critical points are x = -1, 2
(x - 2) - | - | +
(x + 1) - | + | +
Quotient + | - | +
-------+-----------------+------
-1 0 1 2
Then, the inequality is true in the intervals: (-∞, -1), [2, +∞)
Note that at the point x = -1 is not included in the first interval.
Example, solve:
(x - 1)2(x + 2)
———————————————— < 0,
x - 3
The critical points are x = -2, 1, 3
(x - 3) - | - | - | +
(x - 1)2 + | + | + | +
(x + 1) - | + | + | +
Quotient + | - | - | +
-------+-----------------+-----------+------
-2 -1 0 1 2 3
Then, the inequality is true in the open interval: (-2, 3) Note that at the term
(x - 1)2 is always positive; the point x = 1 does not bound the
solution interval.