Given that `-1 le v le 1, -2 le u le -0.5` and `-2 le z le -0.5` and `w = (v z)/(u)`, then which of the following is necessarily true ?

Options

A. A) `-0.5 le w le 2`
B. B) `-4 le w le 4`
C. C) `-4 le w le 2`
D. D) `-2 le w le -0.5`

Answer

To solve the problem, we need to find the range of the variable \( w \) given the expressions and inequalities provided. Let's break it down step by step.

### Step 1: Identify the given inequalities
We have the following inequalities:
- \( -1 \leq v \leq 1 \)
- \( -2 \leq u \leq -0.5 \)
- \( -2 \leq z \leq -0.5 \)

### Step 2: Write the expression for \( w \)
The expression for \( w \) is given as:
\[ w = \frac{vz}{u} \]

### Step 3: Determine the extreme values of \( v \), \( z \), and \( u \)
To find the minimum and maximum values of \( w \), we need to consider the extreme values of \( v \), \( z \), and \( u \).

### Step 4: Finding the minimum value of \( w \)
To find the minimum value of \( w \), we will take:
- \( v = -1 \) (minimum value of \( v \))
- \( z = -2 \) (minimum value of \( z \))
- \( u = -0.5 \) (maximum value of \( u \))

Now substituting these values into the expression for \( w \):
\[ w = \frac{(-1)(-2)}{-0.5} = \frac{2}{-0.5} = -4 \]

### Step 5: Finding the maximum value of \( w \)
To find the maximum value of \( w \), we will take:
- \( v = 1 \) (maximum value of \( v \))
- \( z = -2 \) (minimum value of \( z \))
- \( u = -0.5 \) (maximum value of \( u \))

Now substituting these values into the expression for \( w \):
\[ w = \frac{(1)(-2)}{-0.5} = \frac{-2}{-0.5} = 4 \]

### Step 6: Conclusion on the range of \( w \)
From the calculations:
- The minimum value of \( w \) is \( -4 \).
- The maximum value of \( w \) is \( 4 \).

Thus, we can conclude that:
\[ -4 \leq w \leq 4 \]

### Step 7: Identify the correct option
Based on the range derived, we can now check the options provided in the question. The correct option that reflects this range is:
\[ -4 \leq w \leq 4 \]