(x−4)(x+3)≤0

Sure, let's solve the inequality step by step:

1. First, we need to find the roots of the equation (x-4)(x+3) = 0. This equation is equal to zero when x = 4 or x = -3.

2. Now, we need to test the intervals defined by these roots on the number line. The intervals are (-∞, -3), (-3, 4), and (4, ∞).

3. Choose a test point in each interval and substitute it into the inequality.

   For interval (-∞, -3), let's choose x = -4. Substituting -4 into the inequality gives (-4-4)(-4+3) = (-8)(-1) = 8 which is greater than 0. So, this interval does not satisfy the inequality.

   For interval (-3, 4), let's choose x = 0. Substituting 0 into the inequality gives (0-4)(0+3) = (-4)(3) = -12 which is less than 0. So, this interval satisfies the inequality.

   For interval (4, ∞), let's choose x = 5. Substituting 5 into the inequality gives (5-4)(5+3) = (1)(8) = 8 which is greater than 0. So, this interval does not satisfy the inequality.

4. Therefore, the solution to the inequality (x-4)(x+3) ≤ 0 is -3 ≤ x ≤ 4.