Solving Rational Inequalities Worksheet:
Worksheet given in this section will be much useful for the students who would like to practice solving rational inequalities.
Problem 1 :
Find all values of x for which
x^{3}(x − 1)/(x − 2) > 0
Problem 2 :
Find all values of x for which
(2x - 3)/(x - 2) (x - 4) < 0
Problem 3 :
Find all values of x for which
(x^{2} - 4)/(x^{2} - 2x - 15) ≤ 0
Problem 1 :
Find all values of x for which
x^{3}(x − 1)/(x − 2) > 0
Solution :
x^{3}(x − 1)/(x − 2) > 0
Thus, x^{3}(x − 1) and (x − 2) are both positive or both negative.
So let us find out the signs of x^{3}(x − 1) and x - 2 as follows
Values of x |
x^{3} |
x−1 |
x-2 |
x^{3}(x−1)/(x−2) |
x < 0 |
- |
- |
- |
- |
0 < x < 1 |
+ |
- |
- |
+ |
1 < x < 2 |
+ |
+ |
- |
- |
x > 2 |
+ |
+ |
+ |
+ |
The values from the intervals (0, 1) and (1, 2) satisfies the above rational inequality.
Hence, the solution is (0, 1) U (1, 2).
Problem 2 :
Find all values of x for which
(2x - 3)/(x - 2) (x - 4) < 0
Solution :
(2x - 3)/(x - 2) (x - 4) < 0
Let f(x) = (2x - 3)/(x - 2) (x - 4)
Thus, (2x - 3) and (x−2) (x - 4) are both positive or both negative.
So let us find out the signs of (2x - 3) and (x−2) (x-4) as follows
Values of x |
2x-3 |
(x-2)(x-4) |
f(x) |
x < 3/2 |
- |
+ |
- |
3/2 < x < 2 |
+ |
+ |
+ |
2 < x < 4 |
+ |
- |
- |
x > 4 |
+ |
+ |
+ |
The values from the intervals (-∞, 3/2) and (2, 4) satisfies the above rational inequality.
Hence, the solution is (-∞, 3/2) U (2, 4).
Problem 3 :
Find all values of x for which
(x^{2} - 4)/(x^{2} - 2x - 15) ≤ 0
Solution :
(x^{2} - 4)/(x^{2} - 2x - 15) ≤ 0
Let us factorize both numerator and denominator.
(x^{2} - 4)/(x - 5) (x + 3) ≤ 0
Let f(x) = (x + 2) (x - 2)/(x - 5) (x + 3)
Thus, (x + 2), (x - 2), (x - 5) and (x + 3) are both positive or both negative.
So let us find out the signs of (x + 2), (x - 2), (x - 5) and (x + 3) as follows
Values of x |
(x+2) (x-2) |
(x-5) (x+3) |
f(x) |
x < -3 |
+ |
+ |
+ |
-3 < x < -2 |
+ |
- |
- |
-2 < x < 2 |
- |
- |
+ |
2 < x < 5 |
+ |
- |
- |
x > 5 |
+ |
+ |
+ |
The values from the intervals (-3, -2] and [2, 5) satisfies the above rational inequality.
Hence, the solution is (-3, -2] and [2, 5).
After having gone through the stuff given above, we hope that the students would have understood, how to solve rational inequalities.
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