Question

If \(-1 \leq x \leq 2\) and \(1 \leq y \leq 3\), find the least possible value of \(2y - 3x\).

• A. \(-1\)
• B. \(-3\)
• C. 0
• D. \(-4\)

Hint:

To minimize a subtraction like \(2y-3x\), take the smallest value for the term being added and the largest value for the term being subtracted.

Answer 1

The Correct Option is D

Step 1: Understand how each term behaves.
We want the smallest possible value of \(2y - 3x\), where \(y\) ranges over \([1,3]\) and \(x\) ranges over \([-1,2]\).
The term \(2y\) grows as \(y\) grows, so it is smallest when \(y\) is at its smallest value.
The term \(-3x\) shrinks as \(x\) grows (because of the negative sign), so \(-3x\) is smallest when \(x\) is at its largest value.

Step 2: Pick the values that minimize the expression.
To minimize \(2y - 3x\), we take the smallest allowed \(y\), which is \(y = 1\), and the largest allowed \(x\), which is \(x = 2\).

Step 3: Substitute and calculate.
\[ 2y - 3x = 2(1) - 3(2) = 2 - 6 = -4 \]

Step 4: Check that this really is the minimum.
Any other combination, such as \(y=1, x=-1\) giving \(2+3=5\), or \(y=3, x=2\) giving \(6-6=0\), gives a larger (less negative) value, confirming that \(y=1, x=2\) truly gives the least value.

Final Answer:
The least possible value of \(2y - 3x\) is \(-4\). \[ \boxed{-4} \]