Problem 15 - Algebra and Trigonometry with Analytic Geometry

Question

\(\left\\{\begin{aligned} 3 x+y & \leq 6 \\ y-2 x & \geq 1 \\ x & \geq-2 \\\ y & \leq 4 \end{aligned}\right.\)

Step-by-Step Solution

Answer

Graphical overlapping region satisfies all inequalities.

Step 1: Identify the Constraints

The given system consists of four inequalities. \(3x + y \leq 6\), \(y - 2x \geq 1\), \(x \geq -2\), and \(y \leq 4\). These inequalities define a feasible region on the coordinate plane where all conditions are satisfied simultaneously.

Step 2: Graph the First Inequality

First, draw the line for \(3x + y = 6\). To graph this boundary, find the x-intercept by setting \(y = 0\), thus \(3x = 6\) which gives \(x = 2\). Find the y-intercept by setting \(x = 0\) giving \(y = 6\). Since the inequality is \(\leq\), shade the region below this line.

Step 3: Graph the Second Inequality

Draw the line for \(y - 2x = 1\). Find the x-intercept by setting \(y = 0\), thus \(-2x = 1\) which gives \(x = -0.5\). Find the y-intercept by setting \(x = 0\) giving \(y = 1\). Since the inequality is \(\geq\), shade the region above this line.

Step 4: Apply the Additional Constraints

Now, apply the constraints \(x \geq -2\) and \(y \leq 4\) by drawing vertical and horizontal lines at these points respectively. The region satisfying \(x \geq -2\) is to the right of the line, while \(y \leq 4\) is below the y = 4 line.

Step 5: Determine the Feasible Region

The feasible region is the area where all shaded regions from the inequalities overlap on the graph. This region is the set of all points (x, y) that satisfy all the inequalities simultaneously.

Key Concepts

Feasible Region

In a system of inequalities, the feasible region represents the area on the coordinate plane where all the inequalities are satisfied. This is a crucial concept because it visually demonstrates where the solutions to the system exist. Each point within the feasible region is a potential solution that meets all given inequality constraints.
For instance, if our inequalities describe limitations or requirements in a real-world problem, the feasible region illustrates the range of possible solutions. In the exercise provided, the feasible region is the part where all shaded areas from different inequalities intersect. Essentially, this area is bounded on all sides by the lines described by the inequalities, creating a polygon-shaped area on the graph.

To better understand this, remember that:

• Points inside the feasible region satisfy all inequalities.
• Points outside the region do not satisfy at least one inequality.

Graphing Inequalities

Graphing inequalities involves several steps that help visualize the solutions to a system. The first step is to transform each inequality into an equation by replacing inequality signs with equality. This allows us to draw the boundary line of the inequality. For example, from the inequality \(3x + y \leq 6\), we first graph the line \(3x + y = 6\).
When graphing, it’s crucial to remember:

• For \(\leq\) or \(\geq\), use a solid line to indicate that points on the line are included in the solution.
• For \(<\) or \(>\), use a dashed line because points on the line are not included.
• Once the line is drawn, identify which side of the line satisfies the inequality by testing points.

Graphing multiple inequalities on the same coordinate plane allows us to find the feasible region where solutions overlap. This process, though seemingly complex, becomes intuitive with practice.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we plot our inequalities. It's defined by two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). Where these axes intersect is the coordinate plane's origin, designated as (0,0).
Familiarity with the coordinate plane is vital for graphing inequalities because:

• It allows us to plot intercepts, pivotal for defining boundary lines.
• Shading specific regions helps determine which side of the inequality they represent.

For instance, in our exercise, the intersection of the line from \(3x + y = 6\) and the x-axis is the x-intercept, while the intersection with the y-axis is the y-intercept. These intercepts guide us in drawing accurate representations of the inequalities and ultimately help in determining the feasible region most effectively.

Inequality Constraints

Inequality constraints describe limits or conditions in the system of inequalities. Each constraint sets a specific requirement that must be met. For example, \(3x + y \leq 6\) is a constraint that limits the possible values of x and y in the solution.
Understanding inequality constraints is essential because they:

• Define the boundaries of the feasible region.
• Can represent real-world limits, like budget caps or time restrictions in practical applications.

In graphing, inequality constraints tell us not just where the lines are but also which areas we should shade to include or exclude points. This shading indicates which parts of the coordinate plane satisfy the inequality. In our original exercise, after applying each inequality constraint, the overlapping shaded areas indicate where all conditions are simultaneously satisfied, thus defining the feasible region.