From words to inequalities
Many real-life limits are written as linear inequalities in two variables. Read each condition and match a phrase to a symbol: at least / minimum gives $\geq$, at most / maximum gives $\leq$, more than gives $>$, and less than gives $<$. If a shop buys $x$ chairs and $y$ tables, the cost limit RM800 with chairs RM40 and tables RM100 becomes $40x+100y\leq 800$.
Key formula
General form: $ax+by\leq c$ (or with $\geq,<,>$). Non-negative real quantities also need $x\geq 0$ and $y\geq 0$.
Feasible region and solutions
Draw each boundary line $ax+by=c$. Use a solid line for $\leq$ or $\geq$ and a dashed line for $<$ or $>$. Shade the side that satisfies the inequality (test the point $(0,0)$ when it is not on the line). The overlap of all shaded regions is the feasible region; every point in it satisfies all conditions at once.
Worked example
A stall makes $x$ cups of tea and $y$ cups of coffee. It has at most 12 cups total, and coffee is at least twice the tea. Write the inequalities and find the maximum coffee if $x\geq 1$.
Conditions: $x+y\leq 12$, $y\geq 2x$, $x\geq 1$, $y\geq 0$. Testing integers, $x=1$ gives $y\leq 11$ and $y\geq 2$, so max $y=11$. Check: $1+11=12\leq 12$ and $11\geq 2$. Maximum coffee $=11$ cups.
Remember
- Solid line includes the boundary; dashed line excludes it.
- Test $(0,0)$ to decide which side to shade.
- Integer solutions are lattice points inside or on the feasible region.