Log 8/3 * log 1/2 (𝑥2−𝑥−6) ≥ 0

To solve this inequality, we first need to rewrite it in a more simplified form.

Given: log 8/3 * log 1/2 (𝑥^2−𝑥−6) ≥ 0

We can simplify this inequality by using the properties of logarithms.

First, we can rewrite log 8/3 as log(8) - log(3) and log 1/2 as -log(2).

So, the inequality becomes:

(log(8) - log(3)) * (-log(2)) * (𝑥^2−𝑥−6) ≥ 0

Now, we can simplify further by combining the logarithms:

-log(8) + log(3) * log(2) * (𝑥^2−𝑥−6) ≥ 0

Now, we need to solve for 𝑥.

To do this, we need to consider the properties of logarithms.

For a logarithm to be greater than or equal to 0, the argument inside the logarithm must be greater than 1.

So, we need to solve the inequality:

𝑥^2−𝑥−6 > 1

𝑥^2−𝑥−7 > 0

Now, we need to factor the quadratic equation:

(𝑥-3)(𝑥+2) > 0

Now, we need to find the critical points by setting the equation equal to 0:

𝑥-3 = 0 𝑥 = 3

𝑥+2 = 0 𝑥 = -2

Now, we can test the intervals between the critical points to determine when the inequality is true:

For 𝑥 < -2: (𝑥-3) and (𝑥+2) are both negative, so the inequality is false.

For -2 < 𝑥 < 3: (𝑥-3) is negative and (𝑥+2) is positive, so the inequality is true.

For 𝑥 > 3: (𝑥-3) and (𝑥+2) are both positive, so the inequality is true.

Therefore, the solution to the inequality is 𝑥 < -2 or 𝑥 > 3.