Answer: The solution of the given inequality is [tex](-\infty,-1]\cup (7,\infty)[/tex].
Explanation:
The given inequality is,
[tex]\frac{x^2(x+1)^3}{(x-7)(x+3)^2(-x^2-1)} \leq 0[/tex]
The related equation is,
[tex]\frac{x^2(x+1)^3}{(x-7)(x+3)^2(-x^2-1)}=0[/tex]
By Equating each factor equal to 0. we get x = -3, -1, 0, 7. These four points divides the number line in 5 intervals.
[tex](-\infty,-3),(-3,-1],[-1,0],[0,7),(7,\infty)[/tex]
We will not included -3 and 7 because for x=-3,7 the function is not defined.
Choose any point from each interval. if the point satisfies the inequality it means the interval is the solution of the inequality.
The value of expression on the left side of the inequality either positive of negative as show in the figure.
Since the sign of inequality is less than or equal, therefore interval with negative signs ae the solutions of the given inequality.
The inequality [tex](-\infty,-3),(-3,-1]\text{ and } (7,\infty)[/tex] satisfies the inequality, therefore solution of the given inequality is [tex](-\infty,-1]\cup (7,\infty)[/tex].
