Solve Rational Inequalities
x^2-x-2/x^2+5x+6<0
I cant figure this out. Please help.
numerator (x-2)(x+1)
denominator (x+3)(x+2)
numerator/denominator <0 which means the numerator and denominator are of opposite signs.
Case one: X>0
1a 0<x<2 denominator is +, numerator -, so this is valid.
1b 2<x<inf numerator and denomintor are +, inequality is not valid.
numerator (x-2)(x+1)
denominator (x+3)(x+2)
Case two X<0
2a -1 <x<0
numerator is negative, denom is +, so this range is valid for the identity.
2b -2<x<-1
numerator is +, dem is positve, this is an invalid domain.
2c -3<x<-2 numer is +, den is -, so this is a valid region
you do case 2d, -inf <x < -3
To solve the rational inequality (x^2-x-2)/(x^2+5x+6) < 0, we need to find the values of x for which the fraction is negative.
First, factorize the numerator and the denominator:
Numerator: (x-2)(x+1)
Denominator: (x+3)(x+2)
To determine the sign of the fraction, we need to analyze the signs of both the numerator and the denominator. For the fraction to be negative, the numerator and denominator must have opposite signs.
Case 1: x > 0
In this case, the possible intervals for x are:
1a) 0 < x < 2
For this interval, the denominator (x+3)(x+2) is positive, and the numerator (x-2)(x+1) is negative. Therefore, the inequality is valid in this interval.
1b) x > 2
For this interval, both the numerator and denominator are positive, so the inequality is not valid.
Case 2: x < 0
In this case, the possible intervals for x are:
2a) -1 < x < 0
For this interval, the numerator (x-2)(x+1) is negative, and the denominator (x+3)(x+2) is positive. The inequality is valid in this interval.
2b) -2 < x < -1
For this interval, both the numerator and denominator are positive, so the inequality is not valid.
2c) -3 < x < -2
For this interval, the numerator (x-2)(x+1) is positive, and the denominator (x+3)(x+2) is negative. The inequality is valid in this interval.
2d) x < -3
For this interval, both the numerator and denominator are negative, so the inequality is not valid.
Therefore, the solution to the rational inequality is:
0 < x < 2 (interval in Case 1)
-1 < x < 0 (interval in Case 2)
-3 < x < -2 (interval in Case 2)
-∞ < x < -3 is not a valid interval for the inequality.