Problem 39 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The solution is the line \(y = \frac{2}{3}x - 1\) itself.

Step 1: Rewrite Each Inequality

First, rewrite each inequality in slope-intercept form, which is \(y = mx + b\). For the first inequality, \(2x - 3y \leq 3\), we isolate \(y\): \[ 2x - 3y \leq 3 \] Subtract \(2x\) from both sides: \[-3y \leq -2x + 3 \] Divide each term by \(-3\), and reverse the inequality sign: \[ y \geq \frac{2}{3}x - 1 \] For the second inequality, \(3y \leq 2x - 3\), isolate \(y\): Divide each term by \(3\): \[ y \leq \frac{2}{3}x - 1 \] The system of inequalities becomes: \[ y \geq \frac{2}{3}x - 1 \] \[ y \leq \frac{2}{3}x - 1 \]

Step 2: Analyze the System

The inequalities \(y \geq \frac{2}{3}x - 1\) and \(y \leq \frac{2}{3}x - 1\) indicate lines with the same slope, \(\frac{2}{3}\), and y-intercept, \(-1\). These inequalities describe the same line.

Step 3: Graph the Lines

Graph the line \(y = \frac{2}{3}x - 1\) on the coordinate plane. Because the inequalities are both "less than or equal to" and "greater than or equal to," use a solid line (reflecting that points on the line satisfy both inequalities).

Step 4: Shade the Solution Region

Since both inequalities overlap, the solution is exactly the line itself. No need to shade outside the line, as no other area satisfies both inequalities except the line.

Key Concepts

Graphing Linear Inequalities

Graphing linear inequalities involves shading a region on a graph that satisfies the inequality. When graphing, it's crucial to first convert the inequality into a more familiar form, such as slope-intercept form (discussed in the following section). This form allows us to easily determine the slope and y-intercept of the corresponding line. Once the line is graphed, the next step is to decide which side of the line the inequality represents.

• If the inequality symbol is "less than" (\(<\)) or "greater than" (\(>\)), you will use a dashed line to indicate that points on the line are not included in the solution.
• If the inequality has an "equal to" component, signified by "less than or equal to" (\(≤\)) or "greater than or equal to" (\(≥\)), you will use a solid line. Points on the line are part of the solution.

Once the line style is decided, we shade the region representing all solutions to the inequality. This shading helps visualize the set of points that make the inequality true. In some cases, like in the given exercise, both inequalities might represent the same line; hence the solution may simply be the line itself.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) denotes the y-intercept. This form is particularly useful for graphing because it straightforwardly tells us how to plot the line and makes it easy to recognize the components of the graph.

• Slope (\(m\)): The slope indicates the steepness and direction of a line on the graph. A positive slope means the line rises as it goes from left to right, while a negative slope means it falls.
• Y-intercept (\(b\)): This is the point where the line crosses the y-axis. It gives us a starting point to begin drawing the line.

To use the slope-intercept form in graphing inequalities: - First, convert the inequality into slope-intercept form. - Then, identify the slope and y-intercept from the equation. - Plot the y-intercept on the graph first and use the slope to find another point on the line. Finally, draw the line through these points, determining if it will be solid or dashed depending on the inequality type.

Coordinate Plane

The coordinate plane is a two-dimensional space formed by a horizontal line, called the x-axis, and a vertical line, called the y-axis. These lines intersect at the origin, denoted as \((0, 0)\). Together, they create a grid where any point can be described by a pair of numerical coordinates: the first number corresponds to its position along the x-axis, and the second number corresponds to its position along the y-axis.When graphing on a coordinate plane:

• The x-axis and y-axis divide the plane into four quadrants. These quadrants are typically numbered using Roman numerals, starting from the top right and moving counterclockwise. Each quadrant has distinct combinations of positive and negative values.
• Understanding how to navigate and plot points in each quadrant can significantly simplify analyzing and solving systems of inequalities.

In using the coordinate plane to graph inequalities, plot any lines as stipulated by the slope-intercept form. Then, utilize the axes and the grid to assess regions satisfying one or more inequalities. It's essential to grasp this to correctly determine intersections of inequalities and accurately illustrate the regions representing solutions.