Questions: Find the domain and solve, (a) log(0.5)(3x-2)<-1

Solution Steps

To find the domain of the logarithmic function \(\log_{0.5}(3x-2)\), we need to ensure that the argument of the logarithm is positive. Thus, we solve the inequality: \[ 3x - 2 > 0 \] This simplifies to: \[ x > \frac{2}{3} \] Therefore, the domain is \(x \in \left(\frac{2}{3}, \infty\right)\).

Next, we solve the inequality \(\log_{0.5}(3x-2) < -1\). Converting this logarithmic inequality to its exponential form gives: \[ 3x - 2 < 0.5^{-1} \] Calculating \(0.5^{-1}\) yields \(2\). Thus, we have: \[ 3x - 2 < 2 \] Solving this inequality results in: \[ 3x < 4 \quad \Rightarrow \quad x < \frac{4}{3} \]

Now, we combine the results from the domain and the solution of the inequality. The solution must satisfy both conditions:

1. \(x > \frac{2}{3}\)
2. \(x < \frac{4}{3}\)

Thus, the combined solution is: \[ \frac{2}{3} < x < \frac{4}{3} \]

Final Answer