SOLVING RATIONAL INEQUALITIES WORKSHEET

Problem 1 :

Find all values of x for which

x³(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

(x² - 4)/(x² - 2x - 15) ≤ 0

Detailed Answer Key

Problem 1 :

Find all values of x for which

x³(x − 1)/(x − 2) > 0

Solution :

x³(x − 1)/(x − 2) > 0

Thus, x³(x − 1) and (x − 2) are both positive or both negative.

So let us find out the signs of x³(x − 1) and x - 2 as follows

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

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

(x² - 4)/(x² - 2x - 15) ≤ 0

Solution :

(x² - 4)/(x² - 2x - 15) ≤ 0

Let us factorize both numerator and denominator.

(x² - 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

The values from the intervals (-3, -2] and [2, 5) satisfies the above rational inequality.

Hence, the solution is (-3, -2] and [2, 5).

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