Question

What is the area of the region described by the system of linear inequalities \(x \leq 3, y \leq 1,\) and \(x+y \geq 0 ?\)

Short answer

The area of the region described by the system of linear inequalities is 1.5 square units.

Step-by-step solution

Graph the Individual Inequalities

Begin by graphing each inequality independently. The inequality \(x \leq 3\) creates a vertical line at \(x = 3\) and includes all points to the left of this line. The inequality \(y \leq 1\) creates a horizontal line at \(y = 1\) and includes all points below this line. The inequality \(x + y \geq 0\) creates a line through the origin (0,0) with a slope of -1 and includes all points above and to the right of this line.

Find the Intersection

Now, find the region that satisfies all three inequalities simultaneously. This region is the intersection of the regions formed by each inequality. In this case, the region is a triangle formed by the intersection of the lines \(y = 1\), \(x = 3\), and \(x + y = 0\).

Compute the Area

The vertices of this triangle are at the points (0, 0), (3, 0), and (0, 1). Therefore, the base of the triangle is 3 and the height is 1. Using the formula for the area of a triangle, \(A = 0.5 \times base \times height\), the area of this triangle is \(A = 0.5 \times 3 \times 1 = 1.5\) square units.

Key concepts

Graphing Linear Inequalities

Graphing linear inequalities is crucial in understanding and visualizing the possible solutions to an inequality. Unlike graphing a linear equation, which identifies a line that represents only exact solutions, an inequality includes a range of solutions. Here's the process simplified:

Take the inequality and rearrange it, if necessary, until it is in a standard form, such as \(y \leq mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Next, draw the boundary, which is the line that represents the 'equal to' part of the inequality, by using the slope-intercept or intercept method. Finally, decide which side of the line represents the true solution by choosing a test point not on the line. If the test point satisfies the inequality, then the area on the same side of the line to which the test point belongs is the solution set. Typically, the solution area is shaded on the graph. This method is particularly effective for visual learners as it clearly defines the region of interest.

Systems of Inequalities

When dealing with a system of inequalities, you're looking for the area common to all the inequalities in the system, known as the feasible region. To solve these, first graph each inequality on the same coordinate system. The steps are similar to those for individual inequalities: draw the boundary lines and decide which side of each line satisfies each inequality.

Once all inequalities are graphed, the overlapping shaded region that satisfies all inequalities is the solution to the system. Remember, it can take various shapes, like a polygon or a bounded area. Analyzing how these intersecting regions take shape is key for students to understand the solutions to such systems, going beyond memorizing steps and encouraging critical thinking and spatial awareness.

Area Calculation

Calculating the area of a region formed by linear inequalities is less about applying a formula and more about understanding the shape of the region. After graphing and identifying the shape, you can often decompose the region into basic geometric shapes whose areas are easier to find. For simple figures, such as rectangles and triangles, the formulae are straightforward, and you can obtain the total area by summing up the areas of each shape.

For rectangles, multiply base by height. For triangles, use \(0.5 \times base \times height\). The concept of decomposing complex figures into simpler ones is a powerful strategy in geometry that aids in grasping more intricate area problems by breaking them down into more manageable components.

Triangle Area

A specific case of area calculation is finding the area of a triangle, which is a common outcome when graphing systems of inequalities. To calculate the area of a triangle, you use the formula \(A = 0.5 \times base \times height\).

The 'base' and 'height' must be perpendicular to each other. Identifying the base and height on a coordinate grid usually involves checking the distance between points along the x and y-axes, respectively. Students should be encouraged to practice with coordinates to become adept at spotting the base and height in various orientations and not just when they are parallel to the axes. Understanding the formula's components facilitates the extension of the concept to three-dimensional shapes and further mathematical explorations.