If a, b and c are three real numbers such that `a - b+c lt 1, a + b + c gt -1` and `9a + 3b + c lt -4` , then what is the sign of 'a' ?

Options

A. `+ve` only
B. `-ve` only
C. `+ve` or -ve
D. Cannot be said

Answer

To determine the sign of 'a' given the inequalities involving 'a', 'b', and 'c', we will analyze the conditions step by step.

### Step 1: Write down the inequalities
We have the following inequalities:
1. \( a - b + c < 1 \) (Inequality 1)
2. \( a + b + c > -1 \) (Inequality 2)
3. \( 9a + 3b + c < -4 \) (Inequality 3)

### Step 2: Rearrange the inequalities
From Inequality 1:
\[ a + c < 1 + b \quad \text{(Rearranging gives us a relationship between a, b, and c)} \]

From Inequality 2:
\[ a + c > -1 - b \]

### Step 3: Combine the inequalities
Now we have:
\[ -1 - b < a + c < 1 + b \]
This means that \( a + c \) is bounded by two expressions involving \( b \).

### Step 4: Analyze the third inequality
From Inequality 3:
\[ 9a + 3b + c < -4 \]
We can express \( c \) in terms of \( a \) and \( b \):
\[ c < -4 - 9a - 3b \]

### Step 5: Substitute \( c \) in the combined inequalities
Substituting \( c \) from Inequality 3 into the combined inequalities:
\[ -1 - b < a + (-4 - 9a - 3b) < 1 + b \]
This simplifies to:
\[ -1 - b < -8a - 4 - 4b < 1 + b \]

### Step 6: Solve the inequalities
Now we will solve the left part:
\[ -1 - b < -8a - 4 - 4b \]
Rearranging gives:
\[ 8a < -3 + 3b \quad \Rightarrow \quad a < \frac{-3 + 3b}{8} \]

Now for the right part:
\[ -8a - 4 - 4b < 1 + b \]
Rearranging gives:
\[ -8a < 5 + 5b \quad \Rightarrow \quad a > \frac{-5 - 5b}{8} \]

### Step 7: Analyze the bounds for 'a'
From the inequalities, we have:
\[ \frac{-5 - 5b}{8} < a < \frac{-3 + 3b}{8} \]

### Step 8: Determine the sign of 'a'
To find the sign of 'a', we need to analyze the bounds:
1. If \( b \) is negative, \( \frac{-5 - 5b}{8} \) can be positive, leading to a possibility that \( a \) can also be positive.
2. If \( b \) is positive, \( \frac{-3 + 3b}{8} \) can also be positive.

Thus, depending on the value of \( b \), 'a' can be either positive or negative.

### Conclusion
The sign of 'a' cannot be definitively determined as it can be both positive and negative depending on the values of 'b' and 'c'.