Problem 12 - Algebra A Combined Function

Question

Graph the solutions of each system of linear inequalities $$ \left\\{\begin{array}{l} x \geq-3 \\ y \geq-2 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution region is the area above \( y = -2 \) and to the right of \( x = -3 \).

Step 1: Identify the Inequalities

The given system of inequalities is \( x \geq -3 \) and \( y \geq -2 \). These inequalities describe regions on the coordinate plane.

Step 2: Plot the Boundary Lines

Graph the lines \( x = -3 \) and \( y = -2 \) on the coordinate plane. Use solid lines because the inequalities are non-strict (\( \geq \) sign), indicating that points on the lines satisfy the inequalities.

Step 3: Determine the Solution Region for \( x \geq -3 \)

Shade the region to the right of the line \( x = -3 \) because \( x \geq -3 \) includes all points with an x-coordinate greater than or equal to \(-3\).

Step 4: Determine the Solution Region for \( y \geq -2 \)

Shade the region above the line \( y = -2 \) as \( y \geq -2 \) includes all points with a y-coordinate greater than or equal to \(-2\).

Step 5: Identify the Intersection of the Regions

The solution to the system of inequalities is the overlapping region that satisfies both inequalities. This is the region to the right of \( x = -3 \) and above \( y = -2 \).

Key Concepts

Graphing Inequalities

Graphing linear inequalities is a key method for visualizing solutions in algebra. Unlike equations, which give a specific line or curve on a graph, inequalities show a region where an infinite number of solutions exist. To graph an inequality, start by drawing the boundary line that represents the equation of the inequality. In our exercise, these boundaries are given by the lines \( x = -3 \) and \( y = -2 \). Since the inequalities are "greater than or equal to" (\( \geq \)), we use solid lines. These lines indicate that every point on them is part of the solution.

Once the boundary line is established, the next step involves shading the region that represents all possible solutions. This shading is based on the set of points that satisfy the inequality, making it clear where the solutions lie. Graphing inequalities provides a visual representation that helps in understanding the scope and limitations of the inequality system.

Coordinate Plane

The coordinate plane, often called a Cartesian plane, is a two-dimensional surface where we can graph equations and inequalities. It comprises two number lines that intersect at right angles: the horizontal x-axis and the vertical y-axis. The point where they meet is known as the origin, typically labeled as (0,0). Each location on the plane corresponds to an (x, y) coordinate, which helps in identifying points easily.

When graphing systems of inequalities, like in our exercise, the coordinate plane allows us to precisely draw boundaries and shade regions. The x-coordinate determines a point's horizontal position, while the y-coordinate determines its vertical position. Understanding how to navigate and plot on the coordinate plane is crucial for accurately solving and interpreting systems of inequalities.

Solution Region

The solution region in a graph refers to the area where the inequalities of a system overlap, indicating all possible solutions that satisfy every inequality in the system. For the system provided in this example, \( x \geq -3 \) and \( y \geq -2 \), the solution region is where these two criteria meet on the graph.

To find the solution region, shade the area that satisfies each inequality. For \( x \geq -3 \), shade to the right of the line \( x = -3 \), and for \( y \geq -2 \), shade above the line \( y = -2 \). The solution region is the rectangular area where both shaded regions overlap. This intersection represents all the coordinate points that are solutions to the original system of inequalities.
Identifying the solution region is an essential step as it showcases the combined effect of the inequalities and visually summarizes the set of all feasible solutions.