x^3-x^2-6x greater than or equal to 0
x(x-3)(x+2)greater than or equal to 0
I figured that I have to factor first. Where do I go from here?
To solve the inequality x(x-3)(x+2) ≥ 0, you have already factored the expression correctly. Now, we can use a sign chart to determine the values of x that satisfy the inequality.
1. Create a sign chart by drawing a line and marking three points on it: -2, 0, and 3. These points divide the number line into four intervals.
-2 | 0 | 3
2. Now, choose a test point from each interval and substitute it into the original inequality to determine the sign.
For x < -2, choose x = -3: (-3)(-6)(-1) > 0, which is positive.
For -2 < x < 0, choose x = -1: (-1)(-4)(1) > 0, which is positive.
For 0 < x < 3, choose x = 1: (1)(-2)(3) < 0, which is negative.
For x > 3, choose x = 4: (4)(1)(6) > 0, which is positive.
3. Mark the signs on the sign chart:
-2 (+) | 0 (+) | 3 (-) | +∞ (always positive)
4. Now, we determine the intervals where the inequality is satisfied by looking at the signs:
For (-∞, -2), the sign is positive; thus, this interval does not satisfy the inequality.
For (-2, 0), the sign is also positive; thus, this interval satisfies the inequality.
For (0, 3), the sign is negative; thus, this interval does not satisfy the inequality.
For (3, +∞), the sign is positive; thus, this interval satisfies the inequality.
Therefore, the solution to the inequality x(x-3)(x+2) ≥ 0 is (-2, 0] ∪ (3, +∞).