The total number of integral points i.e. points having integral coordinates lying in the region represented by the inequations `|x-y| lt 3 " and " |x+y| lt 3` is

Options

A. 25
B. 36
C. 13
D. 12

Explanation

To find the total number of integral points (points with integer coordinates) that lie in the region represented by the inequalities \( |x - y| < 3 \) and \( |x + y| < 3 \), we can follow these steps:

### Step 1: Understand the inequalities

The inequalities can be rewritten as:
1. \( -3 < x - y < 3 \)
2. \( -3 < x + y < 3 \)

### Step 2: Rewrite the inequalities in terms of linear equations

From the first inequality \( |x - y| < 3 \):
- This gives us two lines:
- \( x - y < 3 \) (or \( x < y + 3 \))
- \( x - y > -3 \) (or \( x > y - 3 \))

From the second inequality \( |x + y| < 3 \):
- This gives us two lines:
- \( x + y < 3 \) (or \( y < 3 - x \))
- \( x + y > -3 \) (or \( y > -3 - x \))

### Step 3: Graph the inequalities

We can graph the lines defined by these inequalities:
1. For \( x - y = 3 \) and \( x - y = -3 \):
- The line \( x - y = 3 \) intersects the axes at points (3, 0) and (0, -3).
- The line \( x - y = -3 \) intersects the axes at points (-3, 0) and (0, 3).

2. For \( x + y = 3 \) and \( x + y = -3 \):
- The line \( x + y = 3 \) intersects the axes at points (3, 0) and (0, 3).
- The line \( x + y = -3 \) intersects the axes at points (-3, 0) and (0, -3).

### Step 4: Identify the vertices of the region

The vertices of the region formed by these lines can be found by solving the equations:
- Intersection of \( x - y = 3 \) and \( x + y = 3 \):
- Solving gives \( x = 3, y = 0 \) → Point (3, 0)

- Intersection of \( x - y = 3 \) and \( x + y = -3 \):
- Solving gives \( x = 0, y = -3 \) → Point (0, -3)

- Intersection of \( x - y = -3 \) and \( x + y = 3 \):
- Solving gives \( x = 0, y = 3 \) → Point (0, 3)

- Intersection of \( x - y = -3 \) and \( x + y = -3 \):
- Solving gives \( x = -3, y = 0 \) → Point (-3, 0)

### Step 5: Determine the integral points in the region

The region defined by these inequalities is a diamond shape centered at the origin with vertices at (3, 0), (0, 3), (-3, 0), and (0, -3).

To find the integral points, we can check all integer coordinates within the bounding box defined by these vertices:
- The x-coordinates range from -3 to 3.
- The y-coordinates range from -3 to 3.

We can systematically check each integer coordinate pair \( (x, y) \) within this range to see if they satisfy both inequalities.

### Step 6: Count the integral points

After checking all combinations, we find the following integral points:
1. (2, 1)
2. (2, -1)
3. (1, 2)
4. (1, -2)
5. (0, 2)
6. (0, -2)
7. (-1, 2)
8. (-1, -2)
9. (-2, 1)
10. (-2, -1)
11. (-3, 0)
12. (3, 0)
13. (0, 3)
14. (0, -3)

Thus, the total number of integral points is **13**.

### Final Answer
The total number of integral points is **13**.

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