Problem 15 - Intermediate Algebra

Question

Graph the solution set of each system of inequalities. See Example 2 . $$\left\\{\begin{array}{l}2 x+3 y \leq 6 \\\3 x+y \leq 1 \\\x \leq 0\end{array}\right.$$

Step-by-Step Solution

Answer

The solution is the region where all three shaded areas overlap on the graph: below both lines and to the left of the y-axis.

Step 1: Understand the Inequalities

We have three inequalities: \(2x + 3y \leq 6\), \(3x + y \leq 1\), and \(x \leq 0\). These inequalities represent half-planes in a two-dimensional coordinate system. The solution will be the overlapping region that satisfies all three inequalities.

Step 2: Graph the First Inequality

Start by graphing \(2x + 3y \leq 6\). First, draw the line \(2x + 3y = 6\). To do this, find the intercepts: if \(x = 0\), then \(y = 2\); if \(y = 0\), then \(x = 3\). Connect these points. Shade the region below the line because the inequality symbol is \(\leq\).

Step 3: Graph the Second Inequality

Graph \(3x + y \leq 1\) by first mapping the line \(3x + y = 1\). The intercepts are \((x = 0, y = 1)\) and \((x = \frac{1}{3}, y = 0)\). Connect these points and shade the region below the line, which satisfies the inequality.

Step 4: Graph the Third Inequality

For the inequality \(x \leq 0\), shade the area to the left of the vertical line \(x = 0\) (which represents the y-axis).

Step 5: Find the Solution Region

Identify the overlapping shaded region from all the plotted inequalities. This region represents the set of solutions that satisfy all three inequalities.

Key Concepts

Understanding Graphing Inequalities

Graphing inequalities involves translating algebraic inequalities into visual representations on a graph. Each inequality creates a boundary line, dividing the graph into regions that either satisfy or do not satisfy the inequality. For example, with the inequality \(2x + 3y \leq 6\), the boundary line is \(2x + 3y = 6\). This line can be drawn by finding where it crosses the axes, known as the intercepts.

• X-intercept: Set \(y = 0\), resulting in \(x = 3\).
• Y-intercept: Set \(x = 0\), resulting in \(y = 2\).

Connect these points to form the line. Since the inequality symbol is \(\leq\), shade the area below the line where the inequality holds true. This shaded area represents all the possible combinations of \(x\) and \(y\) that satisfy the inequality.
If the inequality were \(<\) instead, the boundary line would typically be dashed, indicating that points on the line are not included in the solution set. Conversely, \(\leq\) or \(\geq\) includes the line as part of the solution.

Exploring the Two-Dimensional Coordinate System

A two-dimensional coordinate system is like a map that helps us visualize and solve systems of inequalities. It consists of two axes: horizontal \(x\)-axis and vertical \(y\)-axis, which intersect at the origin, \((0, 0)\). This system allows plotting of points, lines, and regions. Each point is represented by an ordered pair \((x, y)\).
Lines divide the coordinate plane into two parts. Any line with the form \(ax + by = c\) can be drawn by identifying its intercepts. The regions created by these lines are crucial in determining solutions to inequalities.

• First Quadrant: Both \(x\) and \(y\) are positive.
• Second Quadrant: \(x\) is negative, \(y\) is positive.
• Third Quadrant: Both \(x\) and \(y\) are negative.
• Fourth Quadrant: \(x\) is positive, \(y\) is negative.

By understanding this layout, you can more easily graph lines and identify areas that will be shaded based on the inequalities. This visualization helps pinpoint the solution region where all inequalities overlap.

Identifying the Solution Region

The final step in working with systems of inequalities is to find the solution region on the graph. This region is where all shaded areas from individual inequalities overlap, indicating the set of \((x, y)\) solutions that satisfy every inequality in the system.
First, graph each inequality by drawing its boundary line and shading the appropriate side.

• For \(2x + 3y \leq 6\), shade below the line \(2x + 3y = 6\).
• For \(3x + y \leq 1\), shade below the line \(3x + y = 1\).
• For \(x \leq 0\), shade to the left of \(x = 0\).

The intersection of these shaded regions is the area where all inequalities hold true.
In practice, this overlapping region is often a polygonal area on the graph. Confirm that any point within this area satisfies all three original inequalities. This confirmation ensures that you’ve identified the correct solution region for your system of inequalities.