Trying to solve this rational inequality but can't figure how to factor the numerator. ((3x-2x^2)/(4-x^2))<((3+x)/(2-x))
If we put both sides over a common denominator, we have:
(3x-2x^2)/(40x^2) < (3+x)(2+x)/(4-x^2)
collecting terms, we have (3x^2 + 2x + 6)/(4-x^2) > 0
The numerator is always positive, so as long as the denominator is positive, the inequality holds. We stipulated that x<2, and 4-x^2 > 0 if -2<x<2, so that is the interval where the original inequality holds.
sorry. ignore the phrase about stipulating that x<2.
To solve rational inequalities, you typically start by finding the critical values, which are the values that make the denominator equal to zero. In this case, the denominator of the expression is (4 - x^2) and (2 - x).
To begin, let's find the critical values for the first denominator (4 - x^2):
Setting the expression 4 - x^2 equal to zero:
4 - x^2 = 0
Rearranging the equation:
x^2 - 4 = 0
Now, you can factor the quadratic equation:
(x + 2)(x - 2) = 0
The critical values for the first denominator are x = -2 and x = 2.
Now, let's find the critical value for the second denominator (2 - x):
Setting the expression 2 - x equal to zero:
2 - x = 0
Solving for x:
x = 2
So, the critical value for the second denominator is x = 2.
Now, the critical values for both denominators are x = -2, x = 2.
Next, we need to test different intervals to determine the sign of the inequality. We'll test three intervals: x < -2, -2 < x < 2, and x > 2.
Let's start by testing the interval x < -2:
Choose a test value in this interval, such as x = -3, and substitute it into the inequality:
((3(-3) - 2(-3)^2)/(4-(-3)^2)) < ((3+(-3))/(2-(-3)))
Simplifying the equation:
((-9 - 18)/(4 - 9)) < (0/(2 + 3))
((-27)/(-5)) < 0
Since -27 divided by -5 is a positive number, the inequality holds true.
Next, test the interval -2 < x < 2:
Choose a test value in this interval, such as x = 0, and substitute it into the inequality:
((3(0) - 2(0)^2)/(4-(0)^2)) < ((3+(0))/(2-(0)))
Simplifying the equation:
((0 - 0)/(4 - 0)) < (3/2)
(0/4) < 1.5
Since 0 divided by 4 is zero, and zero is less than 1.5, the inequality holds true.
Finally, test the interval x > 2:
Choose a test value in this interval, such as x = 3, and substitute it into the inequality:
((3(3) - 2(3)^2)/(4-(3)^2)) < ((3+(3))/(2-(3)))
Simplifying the equation:
((9 - 18)/(4 - 9)) < (6/(-1))
((-9)/(-5)) < -6
Since -9 divided by -5 is a positive number, and a positive number is less than a negative number, the inequality does not hold true.
Based on the tests, the solution to the inequality is x < -2 and -2 < x < 2.