Question:
Given that \( -1 \leq v \leq 1, -2 \leq u \leq -0.5 \) and \( -2 \leq z \leq -0.5 \) and \( w = vz/u \), then which of the following is necessarily true?
Given that \( -1 \leq v \leq 1, -2 \leq u \leq -0.5 \) and \( -2 \leq z \leq -0.5 \) and \( w = vz/u \), then which of the following is necessarily true?
Show Hint
The ratio z/u is positive, so first find its smallest and largest values. Then see what multiplying by v, which can be negative, does to that range.
Updated On: Nov 7, 2026
- \( -0.5 \leq w \leq 2 \)
- \( -4 \leq w \leq 4 \)
- \( -4 \leq w \leq 2 \)
- \( -2 \leq w \leq -0.5 \)
Show Solution
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept:
We have to find the smallest and largest values that w can take, then see which option holds for every possible w. Here v, u and z can each vary inside their own interval, so we handle the fraction z/u first and then multiply by v.
Step 2: Key Formula or Approach:
1. If two numbers are both negative, their ratio is positive.
2. For positive numbers, a larger numerator and a smaller denominator give a larger ratio.
Step 3: Range of z/u.
Both z and u are negative, so z/u is positive. The numerator \( |z| \) lies between 0.5 and 2, and the denominator \( |u| \) lies between 0.5 and 2. The smallest ratio is \( 0.5/2 = 0.25 \). The largest ratio is \( 2/0.5 = 4 \). So \[ 0.25 \leq \frac{z}{u} \leq 4 \]
Step 4: Range of w.
Now \( w = v \cdot \frac{z}{u} \), where v is between -1 and 1. Multiplying a number between 0.25 and 4 by v gives a value from \( -1 \times 4 = -4 \) up to \( 1 \times 4 = 4 \). So \[ -4 \leq w \leq 4 \] and both ends can actually be reached, for example v = 1, z = -2, u = -0.5 gives w = 4.
Step 5: Check the options.
Option 1 says \( -0.5 \leq w \leq 2 \). This fails because w = 4 is possible. Option 3 says \( -4 \leq w \leq 2 \). This also fails for w = 4. Option 4 says w is always negative, but v = 1 with z/u positive makes w positive. Option 2 holds for every allowed value.
Final Answer:
w always lies between -4 and 4, which is option 2. \[ \boxed{\text{Option 2: } -4 \leq w \leq 4} \]
We have to find the smallest and largest values that w can take, then see which option holds for every possible w. Here v, u and z can each vary inside their own interval, so we handle the fraction z/u first and then multiply by v.
Step 2: Key Formula or Approach:
1. If two numbers are both negative, their ratio is positive.
2. For positive numbers, a larger numerator and a smaller denominator give a larger ratio.
Step 3: Range of z/u.
Both z and u are negative, so z/u is positive. The numerator \( |z| \) lies between 0.5 and 2, and the denominator \( |u| \) lies between 0.5 and 2. The smallest ratio is \( 0.5/2 = 0.25 \). The largest ratio is \( 2/0.5 = 4 \). So \[ 0.25 \leq \frac{z}{u} \leq 4 \]
Step 4: Range of w.
Now \( w = v \cdot \frac{z}{u} \), where v is between -1 and 1. Multiplying a number between 0.25 and 4 by v gives a value from \( -1 \times 4 = -4 \) up to \( 1 \times 4 = 4 \). So \[ -4 \leq w \leq 4 \] and both ends can actually be reached, for example v = 1, z = -2, u = -0.5 gives w = 4.
Step 5: Check the options.
Option 1 says \( -0.5 \leq w \leq 2 \). This fails because w = 4 is possible. Option 3 says \( -4 \leq w \leq 2 \). This also fails for w = 4. Option 4 says w is always negative, but v = 1 with z/u positive makes w positive. Option 2 holds for every allowed value.
Final Answer:
w always lies between -4 and 4, which is option 2. \[ \boxed{\text{Option 2: } -4 \leq w \leq 4} \]
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