he solution of the rational inequality š(š„) =
š„+1/š„2+š„> 0 is
To solve the rational inequality š(š„) = (š„+1)/(š„^2+š„) > 0, we need to find the values of š„ that satisfy the inequality.
First, let's find the critical points of the rational function. The numerator š„+1 equals 0 when š„ = -1. The denominator š„^2 + š„ cannot equal 0, so we don't have any critical points from the denominator.
Next, let's determine the sign of the rational function in each interval. We can do this by testing a value from each interval in the original inequality.
For š„ < -1, we can choose š„ = -2. Plugging this into the inequality:
š(-2) = (-2+1)/((-2)^2+(-2)) = -1/8 > 0
Since the result is positive, the rational function is positive in this interval.
For -1 < š„ < 0, we can choose š„ = -1/2. Plugging this into the inequality:
š(-1/2) = (-1/2+1)/((-1/2)^2+(-1/2)) = 1/3 > 0
Again, the result is positive, so the rational function is positive in this interval.
For š„ > 0, we can choose š„ = 1. Plugging this into the inequality:
š(1) = (1+1)/(1^2+1) = 2/2 = 1 > 0
The result is positive, indicating the rational function is positive in this interval as well.
Therefore, the solution to the rational inequality š(š„) = (š„+1)/(š„^2+š„) > 0 is š„ < -1 or -1 < š„ < 0, or š„ > 0.