Question

Graph the solution set of the system of inequalities. $$\left\\{\begin{aligned} x+y & \leq 2 \\\\-x+y & \leq 2 \\ y & \geq 0 \end{aligned}\right.$$

Short answer

The graph of the system of inequalities shows an area bounded by the three lines, and shaded area below the lines \(x + y = 2\) and \(-x + y = 2\), and above the line \(y = 0\) is the solution set. The solution set is where all three of these regions overlap.

Step-by-step solution

Graph Each Inequality

Firstly, graph each inequality on the same coordinate plane. Consider each inequality as an equation to get the boundary lines. So graph lines for \(x + y = 2\), \(-x + y = 2\), and \(y = 0\). For \(x + y = 2\), the y-intercept is at \(y = 2\) when \(x = 0\), and the x-intercept is at \(x = 2\) when \(y = 0\). For \(-x + y = 2\), the y-intercept is at \(y = 2\) when \(x = 0\), and the x-intercept is at \(x = -2\) when \(y = 0\). And \(y = 0\) is a horizontal line along the x-axis.

Determine shading area for Each Inequality

Secondly, after drawing the lines, determine which sides of the lines are part of the inequalities' solution. Since the original inequalities were \(x + y \leq 2\), \(-x + y \leq 2\) and \(y \geq 0\), so we shade below line \(x + y = 2\), below line \(-x + y = 2\) and above line \(y = 0\).

Identify the overlapping region

Lastly, identify the region where all shaded regions overlap. That is the area where all three inequalities are satisfied.

Key concepts

Inequality Graphing

Graphing inequalities is a fundamental concept in algebra that involves representing solutions to inequality equations on a coordinate plane. Unlike an equality, where the graph is a single line, an inequality includes a range of values, forming a region. To graph an inequality like \(x + y \leq 2\), we begin by plotting the corresponding line, \(x + y = 2\), which will act as the boundary of the inequality.

The inequality symbol tells us which side of the line is relevant for the solution. For \(\leq\) or \(<\), we shade below the line, whereas for \(\geq\) or \(>\), we shade above the line. With the inequality \(-x + y \leq 2\), we follow the same steps to create its boundary line and shading. When graphing, it's handy to pick test points to determine the correct shading region.

Inequalities such as \(y \geq 0\) are even simpler—the line \(y = 0\) is the x-axis itself, and the solution set includes all points above this horizontal line. To summarize, by graphing the boundary line and applying the correct shading, we visualize the set of all possible solutions to an inequality.

Coordinate Plane

Setting the Stage for Inequalities: The Coordinate Plane

Before proceeding with inequality graphing, a clear understanding of the coordinate plane is essential. This two-dimensional plane is the stage where all the action happens. It's divided into four quadrants by a horizontal line (x-axis) and a vertical line (y-axis).

Each point on this plane has a pair of numbers known as coordinates, denoted as \((x, y)\). The x-value represents the horizontal position, while the y-value indicates the vertical position. When graphing lines and inequalities, identifying intercepts on the axes can be especially helpful. For example, an x-intercept is where the graph crosses the x-axis (where \(y = 0\)), and a y-intercept is where it crosses the y-axis (where \(x = 0\)).

In our exercise, the lines intersect the axes at specific points that help define their positions on the plane, allowing us to visualize and plot the inequalities accurately. Understanding the coordinate plane is fundamental to succeeding in graphing and analyzing inequalities.

Shading Regions

Shading regions is a visual technique used to mark the solution set of an inequality on a graph. When we have a single inequality, shading is quite straightforward. However, systems of inequalities introduce the need to find a common solution area.

After graphing the boundary lines of our inequalities, we move on to this shading step. We shade the region that satisfies each inequality—below for \(\leq\) and above for \(\geq\). The intersection of these shaded areas, or the overlapping region, represents the set of points that satisfy all inequalities simultaneously.

To improve comprehension of this process, consider using different shading patterns (like diagonal lines or dots) for each inequality. The region where the patterns overlap is the solution set we seek. Shading not only helps us visualize the valid area but also ensures we don't consider points that only satisfy one or two inequalities when all must be satisfied together. It's like a Venn diagram for equations, where the overlapping section is our area of interest.