X^3+9x^2-108¡Ý0
If you mean to solve
x^3+9x^2-108 <= 0
first find the factors:
(x+6)^2 (x-3) <= 0
Now, (x+6)^2 >= 0 for all x, so we want
(x-3) <= 0
x <= 3
Makes sense, since if you think of the graph, it comes up from lower left, touches the x-axis at x = -6, dips back down, and finally crosses into positive values at x=3.
To find the solution to the inequality x^3 + 9x^2 - 108 ≥ 0, we need to find the values of x that satisfy the inequality.
Step 1: Factor the equation:
x^3 + 9x^2 - 108 ≥ 0
(x + 12)(x - 3)(x + 3) ≥ 0
Step 2: Find the critical points:
To determine the critical points (where the equation changes sign), set each factor to zero and solve for x:
x + 12 = 0 => x = -12
x - 3 = 0 => x = 3
x + 3 = 0 => x = -3
Step 3: Create a number line:
Now, create a number line and place the critical points on it:
-∞ -12 -3 3 +∞
Step 4: Test the intervals:
Choose a test point from each interval and substitute it into the original inequality to determine whether it satisfies the inequality or not. We can choose test points such as -13, -5, 0, and 4.
For x < -12: Test with -13: (-13 + 12)(-13 - 3)(-13 + 3) ≥ 0 => (-1)(-16)(-10) ≥ 0 => 160 ≥ 0 (True)
For -12 < x < -3: Test with -5: (-5 + 12)(-5 - 3)(-5 + 3) ≥ 0 => (7)(-8)(-2) ≤ 0 => 112 ≤ 0 (False)
For -3 < x < 3: Test with 0: (0 + 12)(0 - 3)(0 + 3) ≥ 0 => (12)(-3)(3) ≥ 0 => -108 ≥ 0 (False)
For x > 3: Test with 4: (4 + 12)(4 - 3)(4 + 3) ≥ 0 => (16)(1)(7) ≥ 0 => 112 ≥ 0 (True)
Step 5: Determine the solution:
By analyzing the test results, we can see that the inequality is true for the intervals x < -12 and x > 3. Therefore, the solution to the inequality x^3 + 9x^2 - 108 ≥ 0 is:
x ≤ -12 or x > 3