91Ó°ÊÓ

The Cartesian plane is like a map for graphing. It helps us understand where everything is placed. This plane is made by two intersecting perpendicular lines. These lines are called axes:

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

This forms a grid where each location is given by a pair of numbers (x, y), called coordinates. The center point where the axes meet is called the origin, valued at (0, 0). In our exercise, graphing on the Cartesian plane lets us easily display equations and inequalities, showing all the possible solutions graphically.

When we set up this plane, we plot all values of x along the horizontal axis and all values of y along the vertical axis. Plotting becomes very straightforward when both axes are drawn, as every coordinate has a clear spot.

Graphing inequalities involves visually showcasing where the solutions to the inequality exist. Let's break it down with "\( y \geq 3 \)":

1. **Plot the boundary line:** For "\( y = 3 \)", this is a horizontal line. All points on this line have a y-value of 3, while x can be any number.
2. **Decide on the line type:** Since "\( y \geq 3 \)" includes the equals part, we use a solid line. A solid line shows both sides of the inequality are included. If it were "\( y > 3 \)", only greater values would be shaded, and the line would be dashed.
3. **Shade the solution area:** Look for all points where the solution is true. Here, it's above the line. Hence, shade the whole area on and above the line to represent every point where y is greater than or equal to 3. Shading makes it easy to visually understand the entire solution.

The solution region is an area on the graph that holds all possible solutions to the inequality. For "\( y \geq 3 \)", we take the following approach:

• The boundary line \( y = 3 \) is part of the solution, drawn as a solid line because the inequality is non-strict.
• To determine the region, remember it involves all y-values above or equal to 3.
• This includes every point on the line, plus those above it. Together, they form the solution region, clearly depicting where the inequality holds true.

This region is shaded, providing an immediate visual cue. Shading is not just for aesthetics; it highlights every possible solution according to the inequality conditions. It's crucial for understanding which parts of the graph are included, offering a clear, graphical representation of algebraic solutions.