Final answer:
To solve the inequality (x+2)(x-4)(x+6)≤0, we determine where the product of these functions is zero or negative by testing intervals around the zeros of the individual factors: x = -6, x = -2, and x = 4. The solution is the union of intervals x ∈ [-6, -2] ∪ [4, ∞).
Step-by-step explanation:
To solve the inequality (x+2)(x−4)(x+6)≤0, we look for the values of x where the product of these factors is either zero or negative. This product will change the sign at the zeros of the individual factors, which are x = -6, x = -2, and x = 4.
Firstly, we need to find when each factor is positive or negative. This gives us intervals we need to test:
- x < -6
- -6 < x < -2
- -2 < x < 4
- x > 4
Select a test point from each interval and substitute it into the inequality to determine if the product is positive or negative in that interval.
By testing, we observe that the product is:
- Negative for x in the interval (-6, -2)
- Positive for x in the interval (-2, 4)
- Negative for x less than -6 or greater than 4
Since we are solving for 'less than or equal to zero', we include the points where the product is zero, i.e., the zeros of the factors.
The solution to the inequality is thus x ∈ [-6, -2] ∪ [4, ∞).