Problem 21 - Intermediate Algebra

Question

Graph the solution set of each system of inequalities on a rectangular coordinate system. $$\left\\{\begin{array}{l}2 x<3 y \\\2 x+3 y \geq 12\end{array}\right.$$

Step-by-Step Solution

Answer

The solution is the overlapping region where shading from both inequalities intersects.

Step 1: Rewrite the Inequalities

First, let's express each inequality in terms of \(y\).For the inequality \(2x < 3y\), divide both sides by 3: \(y > \frac{2x}{3}\).For the inequality \(2x + 3y \geq 12\), isolate \(y\): Subtract \(2x\) from both sides to get \(3y \geq 12 - 2x\), then divide by 3: \(y \geq 4 - \frac{2x}{3}\).

Step 2: Graph the Lines

Convert the inequalities to equality to graph the boundary lines:1. \(y = \frac{2x}{3}\) is the boundary line for \(2x < 3y\).2. \(y = 4 - \frac{2x}{3}\) is the boundary line for \(2x + 3y \geq 12\).Graph each line on the coordinate plane. Use dashed lines for \(y = \frac{2x}{3}\) because its inequality is strict (<), and a solid line for \(y = 4 - \frac{2x}{3}\) because its inequality is inclusive (\(\geq\)).

Step 3: Shade the Solution Region for each Inequality

To determine which side of the line to shade, pick a test point not on the boundary line (often the origin). 1. For \(y > \frac{2x}{3}\), use the point (0,0): Since \(0 > 0\) is false, shade the side not containing (0,0).2. For \(y \geq 4 - \frac{2x}{3}\), use (0,0): Since \(0 \geq 4\) is false, shade the side opposite from (0,0).Shade the appropriate side for each line according to these results.

Step 4: Identify the Overlap

The solution to the system of inequalities is where the shaded regions from both inequalities overlap. Look for the intersection area common to both shaded regions.

Step 5: Conclusion

The overlapping shaded region represents all the coordinate points (x,y) that satisfy both inequalities simultaneously. This is the solution set that should be highlighted on the graph.

Key Concepts

Graphing Inequalities

Graphing inequalities is like graphing equations but with a few important differences. Instead of dealing with one straight line, inequalities will often involve shading a region on the graph.

When graphing an inequality:

• First, convert the inequality into an equation. For example, start by graphing the line of the equation such as from the given inequality.
• Next, determine if your inequality is strict (< or >) or inclusive (≤ or ≥). This distinction informs whether you use a dashed line or a solid line on your graph. Use a dashed line for strict inequalities to indicate that points on the line itself are not included in the solution set.
• Finally, shade the correct side of the line. You can select a test point not on the line to help determine which side to shade. If the inequality holds true with the test point, shade the side containing that point. If not, shade the opposite side.

Graphing inequalities clearly help visualize the solutions to the system and identify where multiple solutions overlap.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is fundamental when graphing equations and inequalities. It's a plane defined by two perpendicular axes: the horizontal (x-axis) and vertical (y-axis).

Key features of the rectangular coordinate system include:

• The x-axis represents horizontal movement and measures the independent variable. This is typically where input values or causes are depicted.
• The y-axis covers vertical movement and measures the dependent variable, relating to outcomes or effects.
• Every point in this plane is described by an ordered pair (x, y), where each pair corresponds to its exact horizontal and vertical position.
• The axes divide the plane into four quadrants, each with its own sign conventions (positive or negative for x and y).

Understanding this system is crucial for graphing as it sets the stage where lines and their relationships are plotted, revealing the nature of various mathematical systems.

Solution Set of Inequalities

The solution set of inequalities contains all points that satisfy all inequalities in a system simultaneously. When represented graphically, it is the region where shaded areas intersect.

To find the solution set:

• First, graph each inequality separately on the same coordinate plane, shading the appropriate regions.
• The intersection of these regions, where the shadings overlap, is the solution set for the system.
• Ensure to highlight this overlapping region clearly, as it visually represents all possible solutions to the given inequalities.
• Always check specific points within this region to verify if they satisfy each inequality, ensuring accuracy.

The comprehension of the solution set is integral for solving systems of inequalities, as it clarifies the constraints and feasible solutions within a defined context.