Final Answer:
The solution to log₉(6x + 27) < 2 is given by [-4.5, 9]. So option C, [-4.5, 9] is correct.
Step-by-step explanation:
The given inequality is log₉(6x + 27) < 2. To solve this logarithmic inequality, we need to rewrite it in exponential form. Recall that logₐ(b) = c can be expressed as aᶜ = b. Applying this to the given inequality:
9² > 6x + 27
Solving for x, we get:
81 > 6x + 27
54 > 6x
x < 9
Now, we must consider the domain of the logarithmic function. The expression inside the logarithm, 6x + 27, must be greater than zero. Setting this inequality:
6x + 27 > 0
6x > -27
x > -4.5
Combining the results, we have -4.5 < x < 9. However, we need to include the endpoints where the expression equals zero and where the logarithm is defined. Therefore, the correct solution is [-4.5, 9].
In conclusion, the solution to log₉(6x + 27) < 2 is represented by the interval [-4.5, 9], taking into account both the inequality derived from the logarithmic expression and the domain restrictions of the logarithmic function.