Problem 33 - Intermediate Algebra

Question

Graph the solution set of each system of inequalities on a rectangular coordinate system. $$\left\\{\begin{array}{l}3 y-5 x<0 \\\5 x-3 y \geq-12\end{array}\right.$$

Step-by-Step Solution

Answer

The solution set is the region below the line \(y = \frac{5}{3}x\) and below or on the line \(y = \frac{5}{3}x + 4\).

Step 1: Rewrite Inequalities in Slope-Intercept Form

The first inequality is \(3y - 5x < 0\), which we can rewrite in slope-intercept form as \(y < \frac{5}{3}x\). The second inequality is \(5x - 3y \geq -12\), which we rewrite as \(y \leq \frac{5}{3}x + 4\). This helps us define the boundary lines to graph as their respective equations \(y = \frac{5}{3}x\) and \(y = \frac{5}{3}x + 4\).

Step 2: Graph the Boundary Lines

First, graph the line \(y = \frac{5}{3}x\). Plot a point at the origin \((0, 0)\), as this is the y-intercept, and use the slope \(\frac{5}{3}\), which means "rise 5, run 3," to plot another point. Draw a dashed line through these points since the inequality is strict \(<\). Next, graph \(y = \frac{5}{3}x + 4\). Start from the y-intercept \((0, 4)\) and use the same slope to plot secondary points. Draw a solid line through these points because the inequality is \(\leq\).

Step 3: Determine the Half-Planes

For \(y < \frac{5}{3}x\), shade the area below its dashed boundary line. This represents all points that satisfy the first inequality. For \(y \leq \frac{5}{3}x + 4\), shade the region below and on the solid line. This represents all points that satisfy the second inequality. The overlapping shaded area will be the solution to the system of inequalities.

Step 4: Identify the Solution Set

The solution set of the system of inequalities is the region where the shaded areas overlap. This is below both lines, where both conditions are satisfied. You can test a point, such as \((0,0)\), to ensure it lies within this region. Substitute \((0,0)\) in both inequalities to see it satisfies those conditions.

Key Concepts

Graphing Inequalities

Understanding how to graph inequalities can help visualize the solution set for a system of inequalities. When working with inequalities, the first step is to express each inequality in slope-intercept form:

• Slope-Intercept Form: A linear equation in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

For instance, with the inequality \( 3y - 5x < 0 \), rearrange it to get \( y < \frac{5}{3}x \). Similarly, convert \( 5x - 3y \geq -12 \) to \( y \leq \frac{5}{3}x + 4 \).
Graph the boundary lines by plotting the y-intercept and using the slope \( \frac{5}{3} \) to identify another point. Draw a dashed line for strict inequalities \(<\) and a solid line for inclusive inequalities \(\leq\). This distinction indicates whether points on the line are included in the solution set.

Rectangular Coordinate System

The rectangular coordinate system is essential for graphing equations and inequalities. It consists of two perpendicular axes:

• The horizontal axis is the x-axis.
• The vertical axis is the y-axis.

These axes intersect at the origin point \((0, 0)\) and divide the plane into four quadrants.
Each point on the plane is represented by an ordered pair \((x, y)\) that corresponds to its position relative to the axes. Identifying the position on this grid will help in plotting points and determining which regions satisfy the inequalities.
By organizing the graph in this way, you can clearly denote boundary lines and shaded regions to visualize the solution set for inequalities.

Solution Set Identification

After graphing the inequalities and their corresponding boundary lines, identify the solution set by determining the overlapping shaded area. Start by shading the region that corresponds to each inequality:

• For \( y < \frac{5}{3}x \), shade the area under the dashed line.
• For \( y \leq \frac{5}{3}x + 4 \), shade the region below or on the solid line.

The solution set is where these two shaded regions overlap. This overlapping area represents all the points that simultaneously satisfy both inequalities.
To confirm the accuracy of this region, choose a test point within the overlap, such as \((0, 0)\), and substitute it into each inequality. If it satisfies both conditions, it confirms the validity of the solution set.