Problem 5 - Elementary Algebra Within Reach

Question

In Exercises \(1-6\), sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{l} x+y \leq 3 \\ x-y \leq 1 \end{array}\right. $$

Step-by-Step Solution

Answer

First, graph each inequality as if it were a normal linear equation. Then, based on the direction of the inequality symbol, shade below the lines formed. The solution to the system is the area where these two shaded regions overlap.

Step 1: Graph each inequality as a regular linear equation

Start by drawing the graph of each corresponding linear equation \(x + y = 3\) and \(x - y = 1\).

Step 2: Determine the solution area for each inequality

For \(x + y \leq 3\), the solutions are all the points on the line \(x + y = 3\) and all the points below this line. So shade the region below the line. For \(x - y \leq 1\), the solutions are all the points on the line \(x - y = 1\) and all the points below this line. So shade the region below this line as well.

Step 3: Determine the solution to the system of inequalities

The solution to the system of inequalities is the area where the shaded regions from Step 2 overlap. This area is defined by the intersection of the solutions of both inequalities.

Key Concepts

Graph of Linear Equations

When approaching a system of linear inequalities, it is essential to first understand the graph of each corresponding linear equation. Linear equations in two variables can be represented as straight lines on a graph. For instance, with the equations \(x + y = 3\) and \(x - y = 1\), we treat them temporarily as equalities. Line graphing proceeds as follows:

• Identify the intercepts or use the slope-intercept form to find points on each line.
• Draw a straight line through these points. Each line divides the coordinate plane into two regions.

This graphical setup is critical, as it forms the basis for the next steps in solving the inequalities.

Solution Area

The solution area of a linear inequality is the set of points that satisfy the inequality. Once you have the graph of a linear equation, determining the solution area involves shading. In our example:

• For the inequality \(x + y \leq 3\), the solution area includes all points on or below the line \(x + y = 3\).
• For the inequality \(x - y \leq 1\), it includes all points on or below the line \(x - y = 1\).

By shading these regions correctly, you depict all possible solutions to the inequalities individually.

Intersection of Inequalities

The intersection of inequalities is where the solutions of multiple inequalities overlap. This overlap represents the set of points satisfying all inequalities simultaneously. To find this intersection:

• Identify where the shaded regions from each inequality cover the same part of the graph.
• These overlapping areas are the solutions to your system of inequalities.

This intersection area is crucial as it visually and mathematically represents the complete solution to the system of linear inequalities at hand.

Shading Regions

Shading regions in a graph of linear inequalities helps in visualizing the solution set. Shading tells us which parts of the graph fulfill the inequality relations. For each inequality:

• Use test points or rules to determine which side of the line to shade.
• Ensure consistent shading direction for inequalities like "less than or equal to," which usually implies shading below the line.

By focusing on accurate shading, you highlight areas of interest, which later assists in identifying the intersection of solutions from multiple inequalities.