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: Treat the problem as checking the corners of a rectangle.
Since $x$ ranges over $[-1,2]$ and $y$ ranges over $[1,3]$ independently, the pair $(x,y)$ can be thought of as a point inside a rectangle with four corners: $(-1,1)$, $(-1,3)$, $(2,1)$, and $(2,3)$.
For an expression like $2y-3x$, which changes steadily (linearly) with $x$ and $y$, the smallest and largest values always occur at one of these four corners, not somewhere in the middle.

Step 2: Evaluate the expression at each corner.
At $(-1,1)$: $2(1) - 3(-1) = 2 + 3 = 5$.
At $(-1,3)$: $2(3) - 3(-1) = 6 + 3 = 9$.
At $(2,1)$: $2(1) - 3(2) = 2 - 6 = -4$.
At $(2,3)$: $2(3) - 3(2) = 6 - 6 = 0$.

Step 3: Pick the smallest value from the list.
Comparing 5, 9, -4 and 0, the smallest is $-4$, occurring at the corner $(x,y) = (2,1)$.

Final Answer:
Checking all four corner combinations confirms the least possible value of $2y-3x$ is $-4$. \[ \boxed{-4} \]