If x an integer greater than -10, but less than 10 and `|x-2|lt3`, then the value of x are

Options

A. `-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4`
B. `0,1,2,3,4,5,6,7,8,9,10`
C. `0,1,2,3,4`
D. `-1,0,1`

Answer

To solve the problem step by step, we need to analyze the given conditions and derive the possible integer values for \( x \).

### Step 1: Understand the given conditions
We have two conditions:
1. \( x \) is an integer such that \( -10 < x < 10 \).
2. \( |x - 2| < 3 \).

### Step 2: Translate the first condition
From the first condition, we can express it as:
\[ -10 < x < 10 \]
This means \( x \) can take any integer value between -9 and 9 inclusive.

### Step 3: Translate the second condition
The second condition involves an absolute value. The expression \( |x - 2| < 3 \) can be rewritten without the absolute value as:
\[ -3 < x - 2 < 3 \]
Now, we can add 2 to all parts of the inequality:
\[ -3 + 2 < x - 2 + 2 < 3 + 2 \]
This simplifies to:
\[ -1 < x < 5 \]

### Step 4: Combine the conditions
Now we have two inequalities:
1. From the first condition: \( -10 < x < 10 \)
2. From the second condition: \( -1 < x < 5 \)

The combined conditions must satisfy both inequalities. The more restrictive condition is:
\[ -1 < x < 5 \]

### Step 5: Identify integer values
Now we need to find the integer values of \( x \) that satisfy \( -1 < x < 5 \). The integers that fit this range are:
\[ 0, 1, 2, 3, 4 \]

### Final Answer
Thus, the integer values of \( x \) that satisfy both conditions are:
\[ \{0, 1, 2, 3, 4\} \]

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