The point (-2,a) lies in the interior of the triangle formed by the lines `y=x,y=-x` and `2x+3y=6` the integral value of a is

Answer

To solve the problem, we need to determine the integral value of \( a \) such that the point \( (-2, a) \) lies inside the triangle formed by the lines \( y = x \), \( y = -x \), and \( 2x + 3y = 6 \).

### Step 1: Find the vertices of the triangle
1. **Intersection of \( y = x \) and \( 2x + 3y = 6 \)**:
- Substitute \( y = x \) into \( 2x + 3y = 6 \):
\[ 2x + 3x = 6 \implies 5x = 6 \implies x = \frac{6}{5} \]
- Therefore, \( y = \frac{6}{5} \). The coordinates are \( \left( \frac{6}{5}, \frac{6}{5} \right) \).

2. **Intersection of \( y = -x \) and \( 2x + 3y = 6 \)**:
- Substitute \( y = -x \) into \( 2x + 3y = 6 \):
\[ 2x + 3(-x) = 6 \implies 2x - 3x = 6 \implies -x = 6 \implies x = -6 \]
- Therefore, \( y = 6 \). The coordinates are \( (-6, 6) \).

3. **Intersection of \( y = x \) and \( y = -x \)**:
- The intersection occurs at the origin \( (0, 0) \).

### Step 2: Identify the vertices of the triangle
The vertices of the triangle formed by the lines are:
- \( A(0, 0) \)
- \( B\left( \frac{6}{5}, \frac{6}{5} \right) \)
- \( C(-6, 6) \)

### Step 3: Determine conditions for the point \( (-2, a) \) to be inside the triangle
To check if the point \( (-2, a) \) is inside the triangle, we can use the concept of inequalities based on the lines forming the triangle.

1. **For line \( AC \) (equation \( y = -x \))**:
- The point \( (-2, a) \) must satisfy:
\[ a < -(-2) \implies a < 2 \]

2. **For line \( AB \) (equation \( y = x \))**:
- The point \( (-2, a) \) must satisfy:
\[ a > -2 \]

3. **For line \( BC \) (equation \( 2x + 3y = 6 \))**:
- Substitute \( (-2, a) \):
\[ 2(-2) + 3a < 6 \implies -4 + 3a < 6 \implies 3a < 10 \implies a < \frac{10}{3} \]

### Step 4: Combine the inequalities
From the inequalities we have:
1. \( a < 2 \)
2. \( a > -2 \)
3. \( a < \frac{10}{3} \)

Combining these, we get:
\[ -2 < a < 2 \]

### Step 5: Find integral values of \( a \)
The integral values of \( a \) that satisfy the inequality \( -2 < a < 2 \) are:
- \( a = -1, 0, 1 \)

### Conclusion
The integral values of \( a \) that satisfy the conditions for the point \( (-2, a) \) to lie inside the triangle are \( -1, 0, 1 \). However, since the question asks for the integral value of \( a \), we can conclude that the possible integral value of \( a \) is:

\[ \text{The integral value of } a \text{ is } 1. \]