If `0 lt x lt 5` and `1 lt y lt 2`, then which of the following is true ?

Options

A. `x + y lt 0`
B. `-3 lt 2x - 3y lt 4`
C. `-6 lt 2x - 3y lt 7`
D. `-3 lt 3x - y lt 2`

Answer

To solve the problem, we need to analyze the inequalities given:

1. We have the inequalities \(0 < x < 5\) and \(1 < y < 2\).
2. We need to find the range for \(x + y\) and other expressions involving \(x\) and \(y\).

### Step 1: Determine the range for \(x + y\)

- The minimum value of \(x + y\) occurs when \(x\) is at its minimum and \(y\) is at its minimum:
\[ \text{Minimum of } x + y = 0 + 1 = 1 \]
- The maximum value of \(x + y\) occurs when \(x\) is at its maximum and \(y\) is at its maximum:
\[ \text{Maximum of } x + y = 5 + 2 = 7 \]

Thus, we can conclude:
\[ 1 < x + y < 7 \]

### Step 2: Analyze the expression \(2x\)

- Since \(0 < x < 5\), we can multiply the entire inequality by 2:
\[ 0 < 2x < 10 \]

### Step 3: Analyze the expression \(-3y\)

- Since \(1 < y < 2\), we multiply the entire inequality by \(-3\). Remember that multiplying by a negative number reverses the inequality:
\[ -6 < -3y < -3 \]

### Step 4: Combine the inequalities for \(2x\) and \(-3y\)

- Now we can combine \(2x\) and \(-3y\):
\[ 0 < 2x < 10 \]
\[ -6 < -3y < -3 \]

When we add these inequalities, we get:
\[ 0 + (-6) < 2x + (-3y) < 10 + (-3) \]
This simplifies to:
\[ -6 < 2x - 3y < 7 \]

### Conclusion

From the analysis, we have determined that:
\[ -6 < 2x - 3y < 7 \]
This means that the expression \(2x - 3y\) lies between -6 and 7.