Problem 24 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The solution is the region in the third quadrant, excluding the line \( y = 0 \).

Step 1: Understand the Inequalities

We have two inequalities given. The first inequality is \( x \leq 0 \), which indicates that the solution set includes all points on or to the left of the vertical line \( x = 0 \). The second inequality is \( y < 0 \), meaning the solution includes all points below the horizontal line \( y = 0 \).

Step 2: Graph the Boundary Line for \( x \leq 0 \)

Draw a vertical line on the Cartesian plane at \( x = 0 \). This line is solid because it includes the points where \( x = 0 \). Shade the entire region to the left of this line, which represents \( x \leq 0 \).

Step 3: Graph the Boundary Line for \( y < 0 \)

Draw a horizontal line at \( y = 0 \). This line is dashed because it does not include the points where \( y = 0 \), indicating \( y < 0 \). Shade the region below this line to represent the inequality \( y < 0 \).

Step 4: Determine the Overlapping Region

Identify the region where the shaded areas from Step 2 and Step 3 overlap. This overlapping region is the solution set for the system of inequalities \( x \leq 0 \) and \( y < 0 \).

Step 5: Verify the Solution Region

Check a point within the overlapping shaded region to ensure it satisfies both inequalities. For instance, the point \((-1, -1)\) is to the left of the line \( x = 0 \) and below the line \( y = 0 \), so it satisfies both \( x \leq 0 \) and \( y < 0 \).

Key Concepts

Graphing Inequalities

Graphing inequalities involves placing inequality equations onto the coordinate plane. This lets us visually determine which parts of the graph satisfy the inequalities. Each inequality proposes a boundary line that divides the coordinate plane into two distinct regions. For example:

• For the inequality \( x \leq 0 \), the boundary line is the vertical line at \( x = 0 \).
• A solid line indicates that points on the line are included in the solution set.
• Whereas, for \( y < 0 \), a dashed line represents that points on the line itself are not included.

To fully solve the problem, you shade the appropriate side of each line where the inequality holds true.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface where every point is described with a pair of numerical coordinates. These coordinates are determined by:

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

Each point is represented as \((x, y)\). For instance, \((-1, -1)\) is found by moving one unit left on the x-axis (since it is negative) and one unit down on the y-axis. When graphing inequalities, this system helps you precisely place boundary lines and identify which regions to shade.

Solution Set

The solution set for a system of inequalities consists of all the points that satisfy all given inequalities. To find this set, you graph each inequality and determine where their shaded regions overlap. Consider the example with inequalities:

• For \( x \leq 0 \), we include the line \( x = 0 \) and all points to its left.
• For \( y < 0 \), the solution is all points below the line \( y = 0 \).

Combining these, the solution set is the intersection of both, often visualized as an overlapping shaded area on the coordinate plane. Ensuring a point like \((-1, -1)\) satisfies both inequalities proves it belongs to the solution set.

Overlapping Region

The overlapping region is the key visual representation of the solution set when graphing systems of inequalities. It is the area on the graph where the shaded parts of each inequality's individual graph intersect. This is crucial because:

• It confirms that all conditions of the system are met in this region.
• Points within this region, such as \((-1, -1)\), should validate both inequalities: \( x \leq 0 \) and \( y < 0 \).

Finding this region involves carefully analyzing where the boundary lines for each inequality in the system overlap, thus simplifying complex mathematical conditions into a tangible visual solution.