`|n-1|lt4`
How many integer n satisfy the inequality above?

Options

A. Two
B. Five
C. Seven
D. Nine

Answer

To solve the inequality \( |n - 1| < 4 \), we can break it down step by step:

### Step 1: Understand the Absolute Value Inequality
The expression \( |n - 1| < 4 \) means that the distance between \( n \) and \( 1 \) is less than \( 4 \). This can be expressed as two inequalities:
\[ -4 < n - 1 < 4 \]

### Step 2: Solve the Compound Inequality
We can split the compound inequality into two parts:
1. \( n - 1 < 4 \)
2. \( n - 1 > -4 \)

### Step 3: Solve Each Inequality
Now, we will solve each part separately.

1. For \( n - 1 < 4 \):
\[ n < 4 + 1 \implies n < 5 \]

2. For \( n - 1 > -4 \):
\[ n > -4 + 1 \implies n > -3 \]

### Step 4: Combine the Results
Combining the results from both inequalities, we have:
\[ -3 < n < 5 \]

### Step 5: Identify the Integer Solutions
Now, we need to find the integer values of \( n \) that satisfy this inequality. The integers that lie between \(-3\) and \(5\) are:
\[ -2, -1, 0, 1, 2, 3, 4 \]

### Step 6: Count the Integer Solutions
Counting these integers, we find:
- \(-2\)
- \(-1\)
- \(0\)
- \(1\)
- \(2\)
- \(3\)
- \(4\)

This gives us a total of **7 integer solutions**.

### Final Answer
Thus, the number of integer \( n \) that satisfy the inequality \( |n - 1| < 4 \) is **7**.

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