Interpret feasible regions in context

A feasible region is the set of ordered pairs that satisfy every inequality in a system simultaneously; in context, each point represents a possible combination of quantities, with its coordinates interpreted using the specified variables and units. The boundary is included or excluded according to the inequality symbol, and the region can be used to identify feasible and infeasible choices and, when relevant, corner points for optimizing a linear objective. The focus is two-variable linear inequalities, excluding nonlinear constraints and higher-dimensional optimization.

Detailed Explanation: Interpret feasible regions in context

A feasible region is the set of all points that satisfy every inequality in a system at the same time. In context, the coordinates of each point represent quantities.

Suppose a bakery makes two products:

$x$ = number of loaves of bread
$y$ = number of batches of muffins

The possible production amounts must satisfy

x+y\le 10

2x+y\ge 8

x\ge 0,\qquad y\ge 0

Step 1: Interpret each inequality

The inequality $x+y\le 10$ means the bakery makes at most $10$ total items.

The inequality $2x+y\ge 8$ means the production combination must provide at least $8$ production units.

The inequalities $x\ge0$ and $y\ge0$ mean negative quantities are not possible.

Step 2: Graph the boundary lines

Replace each inequality with an equation to graph its boundary:

x+y=10

and

2x+y=8

Because both inequalities use $\le$ or $\ge$ , their boundary lines are solid. Points on the lines are included.

Now shade:

Below the line $x+y=10$ , because $x+y$ must be at most $10$ .
Above the line $2x+y=8$ , because $2x+y$ must be at least $8$ .
To the right of the $y$ -axis and above the $x$ -axis, because $x$ and $y$ must be nonnegative.

The overlapping shaded area is the feasible region.

Step 3: Test points

Consider the point $(3,4)$ . This represents $3$ loaves and $4$ batches of muffins.

Check every inequality:

3+4=7\le10

2(3)+4=10\ge8

and both coordinates are nonnegative. Therefore, $(3,4)$ is feasible.

Now consider $(1,2)$ . It fails the second inequality:

2(1)+2=4\not\ge8

Therefore, $(1,2)$ is infeasible, even though it satisfies the first inequality. A point must satisfy all the inequalities to be feasible.

Step 4: Identify corner points, if needed

The vertices, or corner points, of this feasible region are found where boundary lines meet the axes or each other:

On the $y$ -axis, $2x+y=8$ gives $(0,8)$ .
On the $y$ -axis, $x+y=10$ gives $(0,10)$ .
On the $x$ -axis, $2x+y=8$ gives $(4,0)$ .
On the $x$ -axis, $x+y=10$ gives $(10,0)$ .

So the feasible region has corner points

(0,8),\ (0,10),\ (10,0),\ (4,0).

Each point represents a possible production combination. For example, $(10,0)$ means $10$ loaves and $0$ batches of muffins, while $(0,8)$ means $0$ loaves and $8$ batches of muffins.

If an inequality used a strict symbol, such as $x+y<10$ , its boundary would be dashed and points on that line would not be feasible.

Learn by doing: Interpret feasible regions in context

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Two-Variable Linear Inequalities (Pair) - Inequalities to Graph

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