Problem 44 Describe the region that is the ... [FREE SOLUTION]

Chapter 8: Problem 44

Describe the region that is the solution set of each system of inequalities. $$ \begin{array}{l} x \leq 0 \\ y \leq 0 \end{array} $$

Short Answer

Expert verified

The solution set is the third quadrant including the axes.

Step by step solution

- Understand the Inequalities

The system of inequalities given is: 1. $ x \leq 0 $ 2. $ y \leq 0 $. This means we are looking for all points where the x-coordinate is less than or equal to 0 and the y-coordinate is less than or equal to 0.

- Graph the Inequalities

In a coordinate plane, plot the lines $ x = 0 $ and $ y = 0 $. These lines represent the boundaries for the inequalities. Because of the $ \leq $ signs, shade the regions to the left of the line $ x = 0 $ and below the line $ y = 0 $.

- Identify the Intersection

The solution to the system of inequalities is where the shaded regions from each inequality overlap. In this case, the overlapping region is the bottom-left quadrant, including the axes (where both coordinates are $ \leq 0 $).

- Describe the Region

The region that satisfies both inequalities lies in the third quadrant of the coordinate plane, where both x and y are non-positive. This is the region to the left of the y-axis and below the x-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinate Plane

The coordinate plane is essential for graphing systems of inequalities. It's a two-dimensional plane divided by a horizontal x-axis and a vertical y-axis. The point where these axes intersect is called the origin, denoted as (0,0). It divides the plane into four quadrants:

First Quadrant (top-right): Both x and y are positive.
Second Quadrant (top-left): x is negative, y is positive.
Third Quadrant (bottom-left): Both x and y are negative.
Fourth Quadrant (bottom-right): x is positive, y is negative.

Understanding the coordinate plane helps visualize where points and regions lie when solving systems of inequalities. Each point on the plane has coordinates represented as (x, y). These coordinates help identify the location of that point. To start graphing, you plot these points to visualize the solutions to the inequalities.

Graphing Inequalities

Graphing inequalities involves plotting lines and shading regions on the coordinate plane to show all possible solutions. Let's break it down:

First, rewrite each inequality in slope-intercept form (y = mx + b) if needed. For example, x 鈮� 0 and y 鈮� 0 are already simple forms where boundaries are vertical and horizontal lines.
Then, plot the lines that represent the boundaries of the inequalities. For x 鈮� 0, plot a vertical line at x = 0. For y 鈮� 0, plot a horizontal line at y = 0.
Since these inequalities have 'less than or equal to' (鈮�) signs, the boundary lines are solid, indicating points on the line are included in the solution.
Next, shade the appropriate region. For x 鈮� 0, shade to the left of the line x = 0. For y 鈮� 0, shade below the line y = 0. These shaded regions show where the inequalities are met.

Finally, the solution to the system is where the shaded regions overlap. This overlapping part represents all solutions to the system of inequalities. Make sure to clearly mark the shaded regions for clarity.

Solution Set

The solution set of a system of inequalities is the region on the coordinate plane where all inequalities in the system are satisfied simultaneously. Here's how to identify it:

After you graph each inequality on the coordinate plane, look for the regions where the shaded areas overlap. This overlapping region is the solution set.
In the given system, the inequalities are x 鈮� 0 and y 鈮� 0. The graphs of these inequalities shaded the left of the y-axis (for x 鈮� 0) and below the x-axis (for y 鈮� 0).
The region that satisfies both conditions is where both shaded areas overlap. In this case, it's in the third quadrant of the coordinate plane.

This third quadrant region includes all points (x, y) where both coordinates are non-positive. Hence, the solution set can be described as:

All points to the left of the y-axis and below the x-axis, including the axes themselves.
This can be written as {(x, y) | x 鈮� 0 and y 鈮� 0}.

By identifying this overlapping region, you have successfully found the solution set to the system of inequalities.

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91影视

Describe the region that is the solution set of each system of inequalities. $$ \begin{array}{l} x \leq 0 \\ y \leq 0 \end{array} $$

Short Answer

Step by step solution

- Understand the Inequalities

- Graph the Inequalities

- Identify the Intersection

- Describe the Region

Key Concepts

Coordinate Plane

Graphing Inequalities

Solution Set

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