Question

Solve the higher degree inequalities.

x^2(x+5)^3
----------------- ≤0
(3−x)(x+3)^2

Answer 1

The solution of the inequality is
`(-\infty, -5]\cup (3,\infty)`.

Explanation:

The given inequality is

`(x^2(x+5)^3)/((3-x)(x+3)^2)\leq 0`

Equate each factor of numerator and denominator equal to 0.

`x^2=0\Rightarrow x=0`

`(x+5)^3=0\Rightarrow x=-5`

`x+3=0\Rightarrow x=-3`

`x-3=0\Rightarrow x=3`

The points -5,-3, 0 and 3, divides the number line in five parts. The intervals are

`(-\infty, -5], [-5,-3), (-3,0],[0,3), (3,\infty)`

The zero points are -5,-3, 0 and 3 with multiplicity 3, 2, 2 and 1 respectively.

According to the Wavy Curve Method the statement (True or false) change according to the multiplicity of the zeros point.

If the multiplicity is even then statement remains same for next interval and if the multiplicity is odd then statement changes.

Put x=-4

`((-4)^2(1)^3)/((7)(-1)^2)\leq 0`

This statement is false.

From the given number line we can say the solution of the inequality is

`(-\infty, -5]\cup (3,\infty)`

Therefore the solution of the inequality is
`(-\infty, -5]\cup (3,\infty)`.