91Ó°ÊÓ

Graphing inequalities involves creating a visual representation of mathematical conditions on a coordinate plane. Let's break down how to graph the system of inequalities presented in the exercise.

To graph the inequality \( \frac{3}{2}x + y < 3 \), start by rewriting it as an equation: \( y = -1.5x + 3 \). Plot this equation as if it were a line. Since the inequality is 'less than,' represented by '<', you will need to shade the area below this line. This shading indicates all the possible solutions where the inequality holds true.

Next, consider the inequality \( x > 0 \). This inequality represents the set of all points to the right of the vertical line at \( x = 0 \). The line itself acts as a boundary, and since \( x > 0 \), only the right side will be shaded.

The third inequality \( y > 0 \) similarly describes the region above the horizontal line at \( y = 0 \). Shade above this line to indicate regions where \( y \) values are greater than 0.

The challenge in graphing systems of inequalities lies in identifying the correct overlapping region, which is where solutions exist.

The solution set for a system of inequalities includes all the ordered pairs \((x, y)\) that satisfy each inequality in the system. To figure out the solution set, you need to look at the graph.

Here's how you can identify it:

• First, shade the region below the line \( y = -1.5x + 3 \).
• Next, shade the region to the right of the line \( x = 0 \).
• Lastly, shade above the line \( y = 0 \).

After applying the shading, the solution set is revealed as the overlapping area of all these shaded regions. This is the feasible area where all conditions of the system are met simultaneously. In this exercise, this overlapping region is a section in the first quadrant, below the slanted line and above the x-axis.

Visualizing this overlap helps you understand the constraints and possibilities effectively.

The coordinate plane is a two-dimensional surface formed by two number lines intersecting at a right angle. These lines are called axes: the horizontal one is the x-axis, and the vertical one is the y-axis.

Important concepts related to the coordinate plane include:

• The point where the axes intersect is the origin \((0, 0)\).
• Each point on the plane can be described using an ordered pair \((x, y)\).
• The plane is divided into four quadrants. The first quadrant is where both \( x \) and \( y \) are positive.

In the exercise, the focus is primarily on the first quadrant. Because of the given inequalities \( x > 0 \) and \( y > 0 \), only non-negative x and y values are relevant.

The coordinate plane enables you to visualize inequalities and their solution sets clearly. It helps in determining how different linear inequalities interact and where they overlap to provide a feasible solution.