Question

Log3 (x+3) / (x-1) <0

Answer 1

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Answer 2

To solve the inequality $\frac{{\log_3 (x+3)}}{{x-1}} < 0$, we need to find the values of $x$ that satisfy the inequality.

First, note that the inequality is defined when $x > 1$ since the logarithm is not defined for negative values or zero.

To analyze the inequality, we can consider the sign of each factor individually:

1. The numerator, $\log_3 (x+3)$, is positive when $x+3 > 0$, which implies $x > -3$.

2. The denominator, $x-1$, is positive when $x > 1$.

Now let's consider the different intervals for $x$ and determine the sign of the expression:

1. When $x > -3$ and $x > 1$, both the numerator and denominator are positive. Thus, the expression is positive in this interval.

2. When $1 < x < -3$, the numerator is positive, but the denominator is negative. Thus, the expression is negative in this interval.

Therefore, the solution to the inequality is $1 < x < -3$.

In summary, the solution to $\frac{{\log_3 (x+3)}}{{x-1}} < 0$ is $1 < x < -3$.