Constraints: inequalities in two variables

A carpentry workshop makes tables and chairs.
A table takes 3 hours of work, and a chair – one hour.
The workshop’s carpenters have 60 hours of work a week.

How many tables and how many chairs should it make a week? To answer, each quantity to be decided gets a name:

• $\displaystyle x$ – the number of tables a week;
• $\displaystyle y$ – the number of chairs a week.

$\displaystyle x$ and $\displaystyle y$ are the problem’s variables. Each pair $\displaystyle (x,y)$ is a production plan: $\displaystyle (10,20)$ – 10 tables and 20 chairs. The branch of mathematics that looks for the best plan within limits like these is called linear programming.

In the plan $\displaystyle (x,y)$ the work takes $\displaystyle 3x+y$ hours: 3 hours for each table and an hour for each chair.
The carpenters cannot work more than 60 hours, so

A condition like this is called a constraint: a limit every plan must keep, written as an inequality in the two variables.

• “At most”, “no more than” – the sign $\displaystyle \le$.
• “At least”, “no fewer than” – the sign $\displaystyle \ge$.
• The number of products is not negative: $\displaystyle x\ge0$ and $\displaystyle y\ge0$. These are constraints too, in every problem of this kind.

To check a plan, substitute it in the constraint:

On the graph board all the points that satisfy $\displaystyle 3x+y\le60$ are shaded.

• The line $\displaystyle {3x+y}={60}$ is the boundary line: the plans that use all 60 hours. It passes through $\displaystyle (0,60)$ and $\displaystyle (20,0)$.
• Below the line – plans that leave hours free, like $\displaystyle {A}={(10,20)}$.
• Above the line – plans that need more than 60 hours, like $\displaystyle {B}={(15,20)}$.

The reason: the constraint is $\displaystyle y\le60-3x$. For each number of tables $\displaystyle x$, the point on the line is the greatest number of chairs that can be added, and every smaller $\displaystyle y$ fits too. The part of the plane on one side of a line is called a half-plane.

This is how to draw a constraint like $\displaystyle 2x+5y\le100$:

• The boundary line, $\displaystyle {2x+5y}={100}$, passes through two points on the axes: when $\displaystyle {x}={0}$, $\displaystyle {y}={20}$; when $\displaystyle {y}={0}$, $\displaystyle {x}={50}$.
• Choose a point not on the line, usually $\displaystyle (0,0)$, and substitute: $\displaystyle {2\cdot0+5\cdot0}={0\le100}$ – true.
• So shade the side of the line where $\displaystyle (0,0)$ is.

In an inequality with $\displaystyle <$ or $\displaystyle >$ the points on the boundary line are not solutions, so the line is dashed.

A bakery bakes cakes and pies.
A cake takes the oven for 2 hours, and a pie for one hour.
The oven is free for 24 hours a week.
$\displaystyle x$ is the number of cakes a week, and $\displaystyle y$ the number of pies.

Write the constraint of the oven’s hours on the graph board.

Can the bakery bake 8 cakes and 10 pies in a week?
Work out how many oven hours they need, and write the answer.

A half-plane is shaded on the graph board, and the points $\displaystyle A$ and $\displaystyle B$ are on its boundary line.
Write the inequality of the half-plane.

A clothes shop orders shirts and trousers.
A shirt costs the shop 50 ₪, and a pair of trousers 100 ₪.
The budget for the order is at most 3,000 ₪.
The supplier sends an order of at least 20 items.
$\displaystyle x$ is the number of shirts, and $\displaystyle y$ the number of pairs of trousers.

Write the two constraints on the graph board.

Check the plans $\displaystyle (10,15)$ and $\displaystyle (30,20)$: which of them keeps both constraints?