solve the following inequality. write your solution in interval notation.
x^3+9x^2-108 less than or equal to 0
please show all work
Get you started you have to start by guessing
X=3 works
Then do long division
Answer will be x^2+12x+36
Factor
(X+6)(x+6)
3and -6 are your critical values
Now you have to test the different regions
how do I test the different regions??
Just pick a number in the region
Plug back into the function
Neg ans means less than 0
Pos ans means greater than 0
To solve the inequality x^3 + 9x^2 - 108 ≤ 0, we need to find the values of x that satisfy the inequality.
1. Begin by factoring the expression x^3 + 9x^2 - 108 = 0 if possible. Unfortunately, this equation does not easily factor, so we must use an alternative method.
2. Next, we can find the critical points of the expression by setting it equal to zero and solving for x. So, let's solve x^3 + 9x^2 - 108 = 0.
One approach is to use synthetic division or polynomial long division to divide the equation by a factor of (x − a), where "a" is a potential root of the equation. By trying a few potential root values, we find out that x = 3 is a root of the equation x^3 + 9x^2 - 108 = 0.
Performing synthetic division or polynomial long division gives us (x - 3)(x^2 + 12x + 36) = 0.
Now, we can solve the quadratic equation x^2 + 12x + 36 = 0 by factoring it as (x + 6)(x + 6) = 0, which yields two roots, x = -6.
Therefore, the critical points of x^3 + 9x^2 - 108 = 0 are x = 3 and x = -6.
3. Now, we plot these critical points on a number line:
-∞ -6 3 +∞
We need to test a value in each region of the number line to determine if it satisfies the inequality.
4. Let's test a value in the region to the left of -6, like x = -7:
Substitute x = -7 into the expression x^3 + 9x^2 - 108:
(-7)^3 + 9(-7)^2 - 108 = -343 + 441 - 108 = -343 + 333 = -10.
The result is negative (-10 ≤ 0), which means this region satisfies the inequality.
5. Now, test a value within the region between -6 and 3, like x = 0:
Substitute x = 0 into the expression x^3 + 9x^2 - 108:
(0)^3 + 9(0)^2 - 108 = 0 + 0 - 108 = -108.
The result is negative (-108 ≤ 0), so this region also satisfies the inequality.
6. Lastly, test a value in the region to the right of 3, like x = 4:
Substitute x = 4 into the expression x^3 + 9x^2 - 108:
(4)^3 + 9(4)^2 - 108 = 64 + 144 - 108 = 100.
The result is positive (100 > 0), therefore this region does not satisfy the inequality.
7. Based on the testing, the values of x that satisfy the inequality x^3 + 9x^2 - 108 ≤ 0 are the regions:
-∞ < x ≤ -6 and 0 ≤ x ≤ 3.
8. Finally, we write the solution in interval notation:
(-∞, -6] ∪ [0, 3].
Thus, the solution to the inequality x^3 + 9x^2 - 108 ≤ 0, written in interval notation, is (-∞, -6] ∪ [0, 3].