Problem 37 - Precalculus with Limits

Question

In Exercises 33-46, sketch the graph (and label the vertices) of the solution set of the system of inequalities. $$\left\\{\begin{array}{l}{2 x+y>2} \\ {6 x+3 y<2}\end{array}\right.$$

Step-by-Step Solution

Answer

The graph presents two downward-sloping lines. Their intersecting points form the vertices of the feasible region. That region is the solution set for the system of inequalities.

Step 1: Plotting the first inequality

First, plot the line for the first inequality \(2x + y = 2\), which can be rewritten as \(y = -2x + 2\). This is a downward sloping line with a y-intercept of 2 and a slope of -2. The inequality signifies that the solution is above this line, so shade that area.

Step 2: Plotting the second inequality

Next, plot the line for the second inequality \(6x + 3y = 2\), which simplifies to \(2x + y = \frac{2}{3}\), or \(y = -2x + \frac{2}{3}\). This line also slopes downwards but it crosses the y-axis at \(\frac{2}{3}\). The inequality 6x + 3y < 2 signifies that the solution is below this line, so shade this area.

Step 3: Finding the intersection

Identify the region where both the shaded areas from step 1 and step 2 overlap. This common region is the solution to the system of inequalities. It means any point in this region satisfies both inequalities. Also, find the vertices of this region by finding the intersection points of the lines.

Step 4: Label the vertices

Label the vertices by their coordinates. These vertices are the intersection points obtained previously. This completion of the task gives a graphical representation of the solution set for the system of constraints.

Key Concepts

System of Inequalities

A system of inequalities consists of multiple inequalities that share the same variables. When solving such a system, we aim to find all the possible points that satisfy each inequality within the system simultaneously. In the context of graphing, each inequality defines a region on the coordinate plane, and the solution to the system is the region where these individual regions overlap.

For the inequalities given, such as:

• \(2x + y > 2\)
• \(6x + 3y < 2\)

we need to determine where the solutions of these inequalities intersect on a graph. Systems like these are common in linear programming and optimization problems, where the goal is to find the best solution subject to given conditions.

Graphical Representation

Graphical representation is a visual depiction of mathematical concepts on a coordinate plane. In the case of inequalities, we graph the boundaries, often in the form of line equations that result from converting inequalities into equalities.

• For \(2x + y = 2\), the boundary is a straight line that divides the plane into two regions: above or below the line depending on the inequality direction.
• Similarly, \(6x + 3y = 2\) represents the boundary condition for the second inequality.

Once the lines are drawn, the next step is shading. This shading indicates all the points that satisfy the inequality. The lines themselves may or may not be part of the solution, depending on whether the inequality is strict (>) or inclusive (≥). Thus, accurate plotting and shading are crucial for depicting the constraints visually.

Intersection Points

Intersection points are where graphs of lines or curves meet or intersect. In systems of inequalities, these points are often the vertices or corners of the solution region. Finding them involves calculating where two line equations equal each other, which gives us the precise coordinates where the lines intersect.For the system:

• The intersection of \(2x + y = 2\) and \(6x + 3y = 2\) provides the coordinates where both lines meet.

This requires solving the two equations simultaneously to find exact solutions for x and y. The intersection is critical because it helps determine the vertices of the feasible region, specifically when dealing with linear inequalities. Finding these points is necessary to label the vertices on a graph as requested in many exercises, just like ours.

Coordinate Geometry

Coordinate Geometry, or analytic geometry, uses a coordinate system to explore geometrical ideas. With coordinate geometry, we analyze relationships between geometrical shapes using algebra.In this exercise, by graphing lines \(2x + y = 2\) and \(6x + 3y = 2\), we use the slope-intercept form of a line, \(y = mx + b\), to find slopes and intercepts.

• The line \(y = -2x + 2\) has a slope of -2 and y-intersect at (0, 2).
• The line \(y = -2x + \frac{2}{3}\) crosses the y-axis at \(\frac{2}{3}\).

These components help us understand how the lines will interact. Intersections, slopes, and intercepts are fundamental in solving for the intersection points and understanding their graphical representation. Overall, coordinate geometry is essential in solving and visualizing systems of inequalities like this one.