x^3+10x^2+31x <= -30
sorry.
use analytical and graphical methods to solve the inequality.
To solve the inequality x^3 + 10x^2 + 31x <= -30, we can follow these steps:
Step 1: Rewrite the inequality with 0 on one side:
x^3 + 10x^2 + 31x + 30 <= 0
Step 2: Factorize the left side of the inequality if possible. In this case, the expression x^3 + 10x^2 + 31x + 30 cannot be easily factorized. Therefore, we need to use an alternative method.
Step 3: Find the critical points by solving the equation x^3 + 10x^2 + 31x + 30 = 0. This can be done using methods such as factoring or using the Rational Root Theorem.
In this case, solving the equation x^3 + 10x^2 + 31x + 30 = 0 gives us the critical points x = -2, x = -3, and x = -5.
Step 4: Plot the critical points on a number line. It helps to represent them in ascending order: -5, -3, -2.
Step 5: Test a value in each of the regions created on the number line. Choose a value greater than the largest critical point, and a value less than the smallest critical point.
For example, let's test x = -6 (less than -5) and x = 0 (greater than -5).
Step 6: Plug in the tested values back into the inequality x^3 + 10x^2 + 31x + 30 <= 0 and determine if the inequality is satisfied.
For x = -6:
(-6)^3 + 10(-6)^2 + 31(-6) + 30 <= 0
-216 + 360 - 186 + 30 <= 0
-12 <= 0
For x = 0:
0^3 + 10(0)^2 + 31(0) + 30 <= 0
0 + 0 + 0 + 30 <= 0
30 <= 0
Step 7: Analyze the results. We need to find the intervals where the inequality is true. In this case, we notice that the inequality is satisfied for x less than or equal to -5 and x greater than or equal to -3.
Therefore, the solution to the inequality x^3 + 10x^2 + 31x <= -30 is x <= -5 or x >= -3.