91Ó°ÊÓ

Graphing inequalities involves understanding how to visually represent conditions on a graph. When dealing with inequalities such as \(x \leq 2\) and \(y \geq -1\), the goal is to identify the region of the Cartesian plane that satisfies both conditions.

For each inequality, you begin by graphing the corresponding line. However, note the signs:

• A \( \leq \) or \( \geq \) sign means the line itself is included in the solution, so it should be drawn solidly.
• A \( < \) or \( > \) sign would mean using a dashed line, indicating the line itself is not included in the solution set.

Once the lines are drawn, you'll shade the region that satisfies the inequality. The overlapping shaded area from all inequalities in a system will be your solution set.

The Cartesian plane, a crucial tool in graphing inequalities, provides a two-dimensional space where points are determined by a pair of numbers, \(x\) and \(y\).

It is structured by two axes:

• The horizontal axis, known as the x-axis, represents values of the variable \(x\).
• The vertical axis, known as the y-axis, represents values of the variable \(y\).

The intersection of these axes is the origin, denoted as \((0,0)\). Each point on the plane is identified by an ordered pair \((x, y)\). To graph inequalities, you'll plot suitable lines on this plane and shade regions to showcase solutions that meet the conditions set by the inequalities.

The solution set of a system of inequalities is the overlapping region on the Cartesian plane where all conditions are satisfied simultaneously. For the given system \(\{x \leq 2, y \geq -1\}\), it follows these steps:

1. Graph both inequalities: a solid vertical line at \(x = 2\) for \(x \leq 2\) and a solid horizontal line at \(y = -1\) for \(y \geq -1\).
2. Shade the specified areas: left of the line \(x = 2\) and above the line \(y = -1\).
3. Look for the common shaded region, which is the solution set.

This common region is where the solutions to all inequalities intersect. To check any point within this set, substitute its coordinates into each inequality to ensure they meet all conditions.

Understanding vertical and horizontal lines is fundamental when graphing inequalities like \(x \leq 2\) and \(y \geq -1\).

**Vertical Lines**:

• These lines run parallel to the y-axis.
• They are described by equations of the form \(x = a\), where \(a\) is a constant. In this exercise, the line \(x = 2\) is vertical.

**Horizontal Lines**:

• These lines run parallel to the x-axis.
• They are described by equations of the form \(y = b\), where \(b\) is a constant. For this system, the line \(y = -1\) is horizontal.

Each type of line not only helps define boundary conditions in inequalities but also assists in determining the region that represents the solution set.