Question

Graph the feasible region for the following system of inequalities. Tell whether the region is bounded or unbounded. x+4y≤8 3x+4y≥12

Answer 1

The feasible region for the system of inequalities
`\(x + 4y \leq 8\)` and
`\(3x + 4y \geq 12\)` is a bounded area in the coordinate plane. The region is enclosed by the lines x + 4y = 8 and 3x + 4y = 12, and it includes the points of intersection such as (0, 3), (4, 0), and (0, 0). Therefore, the solution is a finite and enclosed area in the coordinate plane.

Step-by-step explanation:

To graph the feasible region for the given system of inequalities, let's start by finding the points of intersection between the two lines formed by the equalities:

1.
`\(x + 4y \leq 8\)`

2.
`\(3x + 4y \geq 12\)`

Let's solve for the points of intersection:

For
`\(x + 4y \leq 8\)`:

Let x = 0:

`\[0 + 4y \leq 8 \implies y \leq 2\]`

Let y = 0:

`\[x + 4 * 0 \leq 8 \implies x \leq 8\]`

So, the points (0, 2), (8, 0), and (0, 0) lie on the line.

For
`\(3x + 4y \geq 12\)`:

Let x = 0:

`\[3 * 0 + 4y \geq 12 \implies y \geq 3\]`

Let y = 0:

`\[3x + 4 * 0 \geq 12 \implies x \geq 4\]`

So, the points (0, 3), (4, 0), and (0, 0) lie on the line.

Now, let's graph these lines on a coordinate system and shade the region that satisfies both inequalities:

```

|\

| \

| \ (0,3)

| \

| \

| \ (4,0)

| \

| \

| \

| \

|__________\

(0,0) (8,0) (0,2)

```

The shaded region above the line
`\(3x + 4y \geq 12\)` and below the line
`\(x + 4y \leq 8\)` is the feasible region.

Bounded or Unbounded:

In this case, the feasible region is bounded because it is a closed region with finite points. It is enclosed by the lines and does not extend infinitely in any direction.