If `0 < ab < ac`, is `a` negative ?
(1) c < 0
(2) b > c

Answer

To determine if \( a \) is negative given the conditions \( 0 < ab < ac \), we will analyze the statements provided step by step.

### Given:
1. \( 0 < ab < ac \)

### We need to find:
Is \( a \) negative?

### Analyzing the given inequality:
From the inequality \( 0 < ab < ac \), we can infer the following:
- Since \( ab > 0 \), both \( a \) and \( b \) must be either both positive or both negative.
- Since \( ac > ab \), we can conclude that \( a(c - b) > 0 \).

### Step 1: Analyze Statement 1
**Statement 1:** \( c < 0 \)

From \( c < 0 \):
- For \( ac > 0 \) (since \( ac > ab > 0 \)), \( a \) must be negative. This is because the product of a negative number \( a \) and a negative number \( c \) will yield a positive product.

Thus, **Statement 1 is sufficient** to conclude that \( a \) is negative.

### Step 2: Analyze Statement 2
**Statement 2:** \( b > c \)

From \( b > c \):
- We know \( ab < ac \) can be rewritten as \( a(b - c) < 0 \).
- Since \( b > c \), it follows that \( b - c > 0 \).
- Therefore, for \( a(b - c) < 0 \), \( a \) must be negative because the product of a positive number \( (b - c) \) and \( a \) is negative only if \( a \) is negative.

Thus, **Statement 2 is also sufficient** to conclude that \( a \) is negative.

### Conclusion:
Both statements independently confirm that \( a \) is negative.

### Final Answer:
Both statements are sufficient to determine that \( a \) is negative.

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