log_((1)/(2))(x+8)-log_((1)/(2))(x-3)gt log_((1)/(2))(3x)

Question

Answer 1

To solve the inequality

, we can use the properties of logarithms.First, let's simplify the left side of the inequality using the quotient rule of logarithms:

So the inequality becomes:

Since the base of the logarithms is

, which is less than 1, the inequality sign will be reversed when taking the logarithm:

Now, let's solve this inequality:

We can solve this quadratic inequality by factoring:

The critical points are

and

. We can test the intervals to find the solution set:For

, the inequality is true.For

, the inequality is false.For

, the inequality is true.Therefore, the solution set for the inequality

is

.

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