Problem 15 - Precalculus: Functions and Graphs

Question

Sketch the graph of the system of Inequalities. $$\left\\{\begin{aligned}3 x+y & \leq 6 \\\y-2 x & \geq 1 \\\x & \geq-2 \\\y & \leq 4\end{aligned}\right.$$

Step-by-Step Solution

Answer

Graph the lines for each inequality and shade the feasibility region to identify the overlap.

Step 1: Convert Inequalities to Equations

First, we convert each inequality into an equation to determine the lines that will define the boundaries of the solution area. Thus, the system of inequalities translates to:\( 3x + y = 6 \), \( y - 2x = 1 \), \( x = -2 \), and \( y = 4 \).

Step 2: Graph the Boundary Lines

Next, graph these lines by finding at least two points for each line:1. For \( 3x + y = 6 \), substitute values for \( x \) to find corresponding \( y \) values. For instance, if \( x = 0 \), then \( y = 6 \). If \( y = 0 \), \( x = 2 \). Draw the line passing through these points.2. For \( y - 2x = 1 \), if \( x = 0 \), then \( y = 1 \). For \( y = 0 \), then \( x = -0.5 \). Draw this line through these points.3. The line \( x = -2 \) is a vertical line at \( x = -2 \).4. The line \( y = 4 \) is a horizontal line at \( y = 4 \).

Step 3: Shade the Feasible Region

Using test points, determine which side of each line to shade:1. For \( 3x + y \leq 6 \): Substitute point \( (0,0) \). Since \( 0 \leq 6 \), shade the region below the line.2. For \( y - 2x \geq 1 \): Using point \( (0,0) \), since \( 0 ot\geq 1 \), shade the region above the line.3. For \( x \geq -2 \): Shade to the right of \( x = -2 \).4. For \( y \leq 4 \): Shade below \( y = 4 \).

Step 4: Identify the Overlapping Area

The solution to the inequalities is the intersection of all the shaded regions identified in the previous step. This is where all the conditions are satisfied simultaneously. Carefully delineate this overlapping region on your graph to highlight the solution area.

Key Concepts

Boundary Lines

When graphing inequalities, understanding boundary lines is essential. Boundary lines are created by converting each inequality in a system to an equation. These equations represent lines on a graph that "bound" the feasible region—the area we need to find where all inequalities are satisfied simultaneously.

For example, consider the inequality system:

• \( 3x + y \leq 6 \) becomes the line \( 3x + y = 6 \)
• \( y - 2x \geq 1 \) becomes the line \( y - 2x = 1 \)
• The inequality \( x \geq -2 \) becomes the vertical line \( x = -2 \)
• Lastly, \( y \leq 4 \) results in the horizontal line \( y = 4 \)

Drawing these boundary lines, usually using at least two points for each, is the first step towards sketching the feasible region. The type of line used (solid or dashed) depends on whether the inequality includes equality ("less than or equal to" or "greater than or equal to"), signaling whether the line itself is part of the solution set.

Feasible Region

The feasible region is the intersection of the solutions for all inequalities in the system. It represents all possible solutions to the inequalities that satisfy every condition simultaneously.

Once the boundary lines are drawn, each inequality divides the plane into two halves. The feasible region will lie on the correct side of each boundary line based on the inequality's sign. If the feasible region is bounded inside specific limits, it often forms a shape like a polygon or region of interest where all conditions are satisfied.

Identifying this region involves understanding which areas of the graph comply with all inequalities simultaneously. Points within this area are solutions to the system, while points outside do not satisfy all the inequalities.

Shading Method

Shading is a graphical method used to identify the solution region on a graph. When dealing with inequalities, we use shading to show which side of a boundary line is included in the solution set.

Here's a simple strategy for shading:

• Pick a test point not on the line, like the origin (0,0), to check if it satisfies the inequality.
• If the inequality holds true for the test point, shade the side that includes the test point. If not, shade the opposite side.

In our current system:

• For \( 3x + y \leq 6 \), the point (0,0) holds true; thus, shade below this line.
• For \( y - 2x \geq 1 \), (0,0) doesn't satisfy the inequality, so shade above the line.
• For \( x \geq -2 \), the solution extends to the right of the line.
• And for \( y \leq 4 \), shade below the line.

The overlapping shaded regions identify the feasible area, where the solution to the system lies.

Intersection of Regions

After shading for each inequality, the final step involves identifying the intersection of regions. This intersection is the actual solution area where all inequalities overlap on the graph.

The intersection of regions forms a distinct shape, often representing the solution set in a visual form. It's essential to highlight this area clearly, as it shows all the permissible combinations of \( x \) and \( y \) that satisfy the entire system of inequalities.

Being able to pinpoint this intersection means you have comprehensively solved the inequality system, allowing for optimization or further analysis based on the specific constraints and requirements set by your equations.