Problem 79 - Algebra 1

Question

GRAPHING Graph the system of linear inequalities. $$ \begin{aligned} &2 x+y \geq 2\\\ &x \leq 2 \end{aligned} $$

Step-by-Step Solution

Answer

The graph of the system of inequalities will depict an overlapped region; above the line \(y = -2x + 2\) and to the left of the line \(x = 2\). This region is the solution set for the system.

Step 1: Draw the Graph of Each Linear Inequality

The first inequality can be rewritten as \(y \geq -2x + 2\). The boundary line has a slope of -2 and a y-intercept of 2. This line should be graphed on a coordinate plane. Similarly, the second inequality can be graphed as a vertical line at \(x = 2\).

Step 2: Identify Solution Regions

For the inequality \(y \geq -2x + 2\), the solution region is above the line, because the \(y\) is greater than the line equation. The solution region for the inequality \(x \leq 2\) is to the left of the vertical line at \(x = 2\).

Step 3: Find Intersection of Solution Regions

The solution of the system of inequalities is the region where the solution regions of both inequalities intersect. This is the area to the left of the line \(x = 2\) and above the line \(y = -2x + 2\).

Step 4: Shade the Solution Region

Shade the overlapping region representing the solution set of the system of inequalities.

Key Concepts

System of Linear Inequalities

When dealing with a system of linear inequalities, you're essentially looking at multiple inequalities that together define a region in the coordinate plane. Each inequality contributes a 'half-plane' where solutions exist. The area where these half-planes overlap is the solution set of the system. This system is a way to address questions that have constraints, such as budget limits or space requirements that need to be satisfied simultaneously.

For example, the inequalities given in the exercise, \(2x + y \geq 2\) and \(x \leq 2\), represent two constraints. The solution set represents all the points \((x,y)\) that satisfy both conditions at the same time. Imagine you're putting together a plan where \(x\) and \(y\) could be resources, and these inequalities symbolize the limits of what you can use or spend on each. Solving a system of linear inequalities is thus like finding the sweet spot where all your constraints are met.

Coordinate Plane

The coordinate plane is a two-dimensional surface where points are determined by a pair of numerical coordinates, which are the values of set ordered pairs \((x, y)\). The plane is divided into four quadrants by the x-axis (horizontal) and y-axis (vertical).

When graphing a system of linear inequalities, the coordinate plane allows us to visually represent the constraints. Each inequality is graphed as a line, and the area on one side of this line contains its solutions. In the given exercise, \(y \geq -2x + 2\) would be a line on this plane, and you'd look at the space above this line for its solutions. Graphing on this plane helps to concretely understand what these abstract equations mean in terms of real, visualizable relationships.

Inequality Graphing

The process of inequality graphing is about visually representing the solutions on a coordinate plane. Unlike equations, inequalities can have multiple, even infinite solutions, and these are represented as regions rather than just lines. Steps to graph an inequality include plotting the 'boundary' line (the associated linear equation), choosing a test point to see where the inequality holds true, and shading the correct side of that boundary line.

In the provided exercise, after graphing the boundary lines for both inequalities, you determine where to shade by selecting test points. Since the inequality \(y \geq -2x + 2\) is satisfied by points above the boundary line, you'd shade upward. For \(x \leq 2\), you'd shade to the left of the vertical line at \(x = 2\). The intersection of these shaded areas is the solution set. It's important to visualize this because it solidifies understanding and makes the concept much easier to grasp in practical terms.