If `(a)/(b)ge(c)/(d)`, then `adlebc` if b and d are of the same sign. Is it true?

Answer

To solve the problem, we need to analyze the given inequality and the conditions provided.

### Step-by-Step Solution:

1. **Given Inequality**:
We start with the inequality:
\[ \frac{a}{b} \geq \frac{c}{d} \]

2. **Cross Multiplication**:
Since \( b \) and \( d \) are of the same sign (both positive or both negative), we can cross-multiply without changing the direction of the inequality:
\[ a \cdot d \geq b \cdot c \]

3. **Case 1: Both \( b \) and \( d \) are Positive**:
If \( b > 0 \) and \( d > 0 \), the inequality remains:
\[ ad \geq bc \]

4. **Case 2: Both \( b \) and \( d \) are Negative**:
If \( b < 0 \) and \( d < 0 \), we can still cross-multiply, but we must remember that multiplying by negative numbers reverses the inequality:
\[ a \cdot d \leq b \cdot c \]
Rearranging gives:
\[ ad \geq bc \]

5. **Conclusion**:
In both cases (whether \( b \) and \( d \) are positive or negative), we find that:
\[ ad \geq bc \]
Thus, the statement \( ad \leq bc \) is incorrect.

### Final Answer:
The statement "If \( \frac{a}{b} \geq \frac{c}{d} \), then \( ad \leq bc \) if \( b \) and \( d \) are of the same sign" is **not true**.