Find the range of z=y/x over the feasible region.

01 / Standard#Your turn

3x+2y\ge13,\quad x\le5,\quad2x-2y+3\ge0

Official paper · jm01-2023 · II.5(b) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Every feasible x lies in [2,5], so division preserves inequalities.

Hint 2

Use lower and upper boundary formulae for y.

Worked solution

For each x, bound the ratio.
$\frac{13}{2x}-\frac32\le\frac yx\le1+\frac{3}{2x},\quad2\le x\le5$
Both bounds decrease with x; extrema occur at C and A.
$z_{\min}=\frac{13}{10}-\frac32=-\frac15,\quad z_{\max}=1+\frac34=\frac74$

−1/5≤z≤7/4.

Checks and common pitfalls: A ratio is not a linear objective; justify the range using x>0 or slopes.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
For each x, bound the ratio.
$\frac{13}{2x}-\frac32\le\frac yx\le1+\frac{3}{2x},\quad2\le x\le5$
Both bounds decrease with x; extrema occur at C and A.
$z_{\min}=\frac{13}{10}-\frac32=-\frac15,\quad z_{\max}=1+\frac34=\frac74$

Think first. Reveal a hint when the class is ready.

2023 JM01