Solve for real ` x: (x + | x + 3 |)/( x + 2) lt 1, x ne - 2 `

Answer

To solve the inequality \((x + |x + 3|)/(x + 2) < 1\) with the condition \(x \neq -2\), we will analyze the expression based on the definition of the absolute value and the sign of \(x + 2\).

### Step 1: Consider the case when \(x + 2 > 0\) (i.e., \(x > -2\))
In this case, we have:
\[ |x + 3| = x + 3 \]
Thus, the inequality becomes:
\[ \frac{x + (x + 3)}{x + 2} < 1 \]
This simplifies to:
\[ \frac{2x + 3}{x + 2} < 1 \]

### Step 2: Multiply both sides by \(x + 2\) (since \(x + 2 > 0\), the inequality sign remains the same)
\[ 2x + 3 < x + 2 \]

### Step 3: Rearrange the inequality
Subtract \(x\) from both sides:
\[ 2x - x + 3 < 2 \]
This simplifies to:
\[ x + 3 < 2 \]
Subtract 3 from both sides:
\[ x < -1 \]

### Step 4: Combine conditions
Since we have \(x > -2\) and \(x < -1\), the solution for this case is:
\[ -2 < x < -1 \]

### Step 5: Consider the case when \(x + 2 < 0\) (i.e., \(x < -2\))
In this case, we have:
\[ |x + 3| = -(x + 3) = -x - 3 \]
Thus, the inequality becomes:
\[ \frac{x - (x + 3)}{x + 2} < 1 \]
This simplifies to:
\[ \frac{-3}{x + 2} < 1 \]

### Step 6: Multiply both sides by \(x + 2\) (since \(x + 2 < 0\), the inequality sign changes)
\[ -3 > x + 2 \]

### Step 7: Rearrange the inequality
Subtract 2 from both sides:
\[ -3 - 2 > x \]
This simplifies to:
\[ -5 > x \quad \text{or} \quad x < -5 \]

### Step 8: Combine conditions
Since we have \(x < -2\) and \(x < -5\), the solution for this case is:
\[ x < -5 \]

### Final Solution
Combining both cases, we have:
\[ (-\infty, -5) \cup (-2, -1) \]