Question

Graph the system of linear inequalities. \(x \geq 0\) \(y \geq 0\) \(x \leq 3\) \(y \leq 5\)

Short answer

The graph of the system of inequalities is a rectangle in the first quadrant of the coordinates system bounded by the lines \(x=0\), \(y=0\), \(x=3\), and \(y=5\).

Step-by-step solution

Graphing x \geq 0

This inequality means x is greater than or equal to 0. On a graph, this is represented by a vertical line at x=0, and the region to the right of this line.

Graphing y \geq 0

This inequality means y is greater than or equal to 0. On a graph, this is represented by a horizontal line at y=0, and everything above this line combines with the area from the previous step.

Graphing x \leq 3

This inequality means x is less than or equal to 3. On a graph, this translates into another vertical line at x = 3. The acceptable area is to the left of this line, which will intersect with the areas from the first two steps.

Graphing y \leq 5

The last inequality is y less than or equal to 5. This corresponds to another horizontal line on the graph at y = 5. The acceptable region is beneath this line. This will overlap with the area determined by the previous inequalities.

Identify the overlapping region

The solution to the system of inequalities is the area of the plane where all the acceptable regions from the previous steps overlap. This would be a rectangle in the first quadrant bounded by the lines \(x=0\), \(y=0\), \(x=3\), and \(y=5\).

Key concepts

System of Linear Inequalities

When dealing with a system of linear inequalities, we're essentially looking at several inequalities that we solve simultaneously. The solution to such a system is the region where the solutions to all the individual inequalities intersect.

For instance, in our exercise, you have four separate inequalities:

• \(x \textgreater\= 0\text{:}\text{} This indicates that x is not less than 0.\text{}\)
• \(y \textgreater\= 0\text{:}\text{} Similarly,\text{} y cannot be less than 0.\text{}\)
• \(x \textless\= 3\text{:}\text{} Here,\text{} x cannot exceed 3.\text{}\)
• \(y \textless\= 5\text{:}\text{} This specifies that y is no more than 5.\text{}\)

Each inequality defines a half-plane on a two-dimensional graph. When graphed, the intersection of these half-planes shows the region where all conditions are satisfied at once. In the given exercise, all four inequalities constrain the solution set to a specific region of the coordinate plane.

Inequality Graphing

Graphing inequalities is a powerful visual way to represent the solution set of an inequality or a system of inequalities. Unlike equations, inequalities do not just have one line or point as a solution; they include a whole area or region.

When graphing an inequality, such as \(x \textgreater\= 0\text{}\text{or}\text{} y \textless\= 5\text{},\text{}\) we start by drawing the 'boundary' line or point. For \(x \textgreater\= 0\text{},\text{}\) the boundary is a vertical line along the y-axis, and for \(y \textless\= 5\text{},\text{}\) it's a horizontal line at the point where y equals 5. These lines are solid because the inequality includes equality (equal to 0 or 5).

If the inequality were strict—such as 'greater than' rather than 'greater than or equal to'—the boundary line would be dashed, indicating that the line itself is not part of the solution. After the boundary is drawn, we shade the appropriate side of each line to show where the inequality holds true. The overlapped shaded region across all graphs represents the solution set to the entire system.

Quadrant System

The quadrant system is fundamental to understanding and graphing inequalities. It divides the two-dimensional plane into four sections, known as quadrants, which are labeled counterclockwise as I, II, III, and IV.

Each quadrant represents a distinct combination of positive and negative values for the x and y coordinates:

• Quadrant I: Both x and y are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both x and y are negative.
• Quadrant IV: x is positive, y is negative.

In our exercise, since all the inequalities involve x and y being greater than or equal to 0, the solution lies entirely within Quadrant I. This region is where both the x and y coordinates of any point will be positive. Understanding this helps in locating the appropriate region for the solution set quickly, without having to process each inequality independently.