`|x+1|gt2`

Answer

To solve the inequality \( |x + 1| > 2 \), we can break it down into two cases based on the definition of absolute value.

### Step 1: Understand the definition of absolute value
The absolute value \( |a| \) can be defined as:
- \( a \) if \( a \geq 0 \)
- \( -a \) if \( a < 0 \)

In our case, we have \( |x + 1| \).

### Step 2: Set up the two cases
We will consider two cases based on the expression inside the absolute value:

**Case 1:** \( x + 1 \geq 0 \) (which means \( x \geq -1 \))
In this case, \( |x + 1| = x + 1 \).

**Case 2:** \( x + 1 < 0 \) (which means \( x < -1 \))
In this case, \( |x + 1| = -(x + 1) = -x - 1 \).

### Step 3: Solve Case 1
For Case 1, we have:
\[ |x + 1| > 2 \implies x + 1 > 2 \]
Subtracting 1 from both sides gives:
\[ x > 1 \]

### Step 4: Solve Case 2
For Case 2, we have:
\[ |x + 1| > 2 \implies -x - 1 > 2 \]
Adding 1 to both sides gives:
\[ -x > 3 \]
Multiplying both sides by -1 (and reversing the inequality) gives:
\[ x < -3 \]

### Step 5: Combine the results
From both cases, we find:
1. From Case 1: \( x > 1 \)
2. From Case 2: \( x < -3 \)

Thus, the solution to the inequality \( |x + 1| > 2 \) is:
\[ x > 1 \quad \text{or} \quad x < -3 \]

### Final Answer:
The solution set is \( x > 1 \) or \( x < -3 \).

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