Discussion

Explanation:

Here we only need to consider the signs of each of these 3 expressions given and not their value.

∴ The range of given inequality will be same as the range for (x + 3)(x - 7)(x + 1) ≤ 0

The critical points here are -3, -1 and 7.

∴ (x + 3)(x - 7)(x + 1) is

> 0 when x > 7 (range not accepted)
< 0 when -1 < x < 7 (range accepted)
> 0 when -3 < x < -1 (range not accepted)
< 0 when x < -3 (range accepted)

∴ For (x + 3)(x - 7)(x + 1) ≤ 0, the acceptable range of x ∈ (-∞, -3] ∪ [-1, 7]

Here we need to be careful about critical points. For x = -1, the original expression is invalid, hence x = -1 should not be included in our range of x.

∴ For $\frac{{(x+3)}^3(x-7)}{(x+1)}$ ≤ 0

x ∈ (-∞, -3] ∪ (-1, 7]

Hence, option (a).

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