Problem 23 - Elementary Algebra Within Reach

Question

In Exercises 23-28, sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{r} -3 x+2 y<6 \\ x-4 y>-2 \\ 2 x+y<3 \end{array}\right. $$

Step-by-Step Solution

Answer

The intersection area of the three inequalities plotted on a graph represents the solution.

Step 1: Rewrite inequalities as equations

Rewrite the inequalities by setting each one equal to 0. Hence, we get the following equations: \(-3x + 2y = 6\), \(x - 4y = -2\), and \(2x + y = 3\). These equations represent the boundary lines of the inequality region.

Step 2: Sketch the lines

Plot the equations on a graph. Generally, express y in terms of x and find two points on each line (for example, when x = 0 and x = 1), then draw the line through these points. Remember to draw solid lines if the inequality is \(<=\) or \(>=\) and dashed lines if the inequality is \(<\) or \(>\). In this case, all lines should be dashed.

Step 3: Determine the area of interest

Identify which side of the lines represent the solution to each inequality. A good method is to choose a test point, usually the origin (0,0) if it does not lie on any of the lines. Substitute the test point into the inequality. If the inequality is true, then shade the side of the line that contains the test point. If the inequality is false, then shade the side that doesn't contain the test point.

Step 4: Find intersection of areas

The solution to the systems of inequalities is the region where the shaded areas from the individual inequalities overlap. Identify and highlight this area.

Key Concepts

Sketching Inequalities Graph

Graphing a system of linear inequalities involves visualizing the solution set on a coordinate plane. The process begins with converting each inequality into its corresponding boundary line by transforming the inequality symbol to an equality symbol. This step is crucial for laying the framework of the graph.

When sketching, you first plot these transformed equations as if they were regular lines. For instance, given an inequality like \( -3x + 2y < 6 \), the corresponding equation for the boundary line is \( -3x + 2y = 6 \). Each boundary line should be carefully drawn according to the original inequality symbol: if the inequality is strict (such as \( < \) or \( > \)), use a dashed line; if it is inclusive (\( \leq \) or \( \geq \)), employ a solid line. This distinction visually communicates which side of the line is part of the solution set.

To identify the valid region for the solution, you will consider the area that satisfies the inequality relative to its boundary line. Each inequality cuts the coordinate plane into two halves, and the solution set of the inequality is the half-plane that satisfies the inequality.

Boundary Lines

Boundary lines play an integral role in the structure of the graph while solving systems of linear inequalities. They are the lines that represent the threshold between the solutions and non-solutions of each inequality when plotted on the graph. A boundary line is derived by taking the inequality equation, such as \( x - 4y > -2 \), and rewriting it as an equation, \( x - 4y = -2 \).

Importance of Boundary Line Representation

The way the boundary line is drawn depends on the inequality's inclusivity; a solid line indicates that points on the line are included in the solution set \( (\geq, \leq) \) whereas a dashed line suggests they are excluded \( (>, <)\). For the exercise provided, all boundary lines should be dashed, conveying that the exact points on the lines are not part of the solution set.

Identifying which side of the boundary line contains the solutions requires testing points or analyzing the inequality. This delineation is the key to visually understanding and solving the inequality, completing the sketch with the correct regions shaded.

Test Point Method

The test point method is a reliable way to determine which half-plane defined by the boundary line represents the solution set to an inequality. To utilize this method, select any point not on the boundary line, and substitute its coordinates into the original inequality. The point \( (0,0) \) often serves as a convenient choice, provided it is not on any of the boundary lines.

When using the test point, if the inequality holds true, then the entire half-plane that includes the test point will be part of the solution set. Conversely, if the inequality does not hold, the opposite half-plane is where the solutions reside. It is a simple yet effective check.

Applying the Test Point Method

Take the inequality \( 2x + y < 3 \), and the point \( (0,0) \) as the test point. Substituting these coordinates yields \( 2(0) + (0) < 3 \) which simplifies to \( 0 < 3 \), a true statement. Consequently, you would shade the side of the boundary line where \( (0,0) \) resides. Extend this check to all inequalities in the system to map out the entire solution region comprehensively. The intersection of all the shaded regions represents the solution to the system of linear inequalities, thus visualizing the feasible solution space.