solve the inequation `(x +1)/(x +3)gt1`

Explanation

To solve the inequation \(\frac{x + 1}{x + 3} > 1\), we will follow these steps:

### Step 1: Rewrite the Inequality
We start with the given inequality:
\[ \frac{x + 1}{x + 3} > 1 \]
Subtract \(1\) from both sides:
\[ \frac{x + 1}{x + 3} - 1 > 0 \]

### Step 2: Combine the Fractions
To combine the fractions, we need a common denominator:
\[ \frac{x + 1 - (x + 3)}{x + 3} > 0 \]
This simplifies to:
\[ \frac{x + 1 - x - 3}{x + 3} > 0 \]
\[ \frac{-2}{x + 3} > 0 \]

### Step 3: Analyze the Inequality
Now we need to determine when \(\frac{-2}{x + 3} > 0\). Since \(-2\) is a negative constant, the fraction will be positive when the denominator is negative:
\[ x + 3 < 0 \]

### Step 4: Solve for \(x\)
Solving the inequality:
\[ x < -3 \]

### Step 5: Write the Solution
The solution to the inequality is:
\[ x \in (-\infty, -3) \]
Note that \(x\) cannot equal \(-3\) since that would make the denominator zero.

### Final Answer
Thus, the solution set is:
\[ x \in (-\infty, -3) \]

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