91Ó°ÊÓ

Graphing inequalities is crucial for visualizing where a certain condition holds true on a coordinate plane. Let's start with the inequality given in the exercise: \( y > x \). This inequality means that the y-values are greater than the x-values in the region we are interested in. To graph this inequality, we first plot the line \( y = x \). This line acts as a boundary and divides the plane into two regions. The line extends diagonally through the origin (0,0) with a slope of 1.
To differentiate these regions, shade the area where the condition \( y > x \) is true.

• Above the line \( y = x \): \( y > x \)
• Below the line \( y = x \): \( y < x \)

Graphing this helps us visually understand where the inequality holds. The region above the line will be shaded and it will not include the line itself because we're dealing strictly with \( > \). This step is essential in problems that involve graphing inequalities.

The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves to understand their relationships. It is divided into four quadrants by the x-axis (horizontal) and y-axis (vertical), intersecting at the origin (0,0). In our exercise, the coordinate plane helps us visualize solutions to the inequality \( y > x \).
Each point on the plane is represented by an ordered pair \((x, y)\), where:

• x is the horizontal distance from the origin
• y is the vertical distance from the origin

Understanding the coordinate plane is vital for graphing and solving inequalities, as it enables us to see where these points lie with respect to the equation or inequality given. When working with inequalities like \( y > x \), graphing on the coordinate plane becomes an intuitive way to grasp which regions satisfy the condition and which do not.

The coordinate plane is divided into four sections called quadrants, each holding unique characteristics. The quadrants are numbered from I to IV in an anti-clockwise manner starting from the upper-right:

• Quadrant I: Positive x and positive y
• Quadrant II: Negative x and positive y
• Quadrant III: Negative x and negative y
• Quadrant IV: Positive x and negative y

From the problem, we know that in Quadrant IV, x-values are positive and y-values are negative. By analyzing the inequality \( y > x \), we find that below the line \( y = x \), the y-values are less than the x-values. Thus, in Quadrant IV, where y is negative and x is positive, \( y < x \), meaning the inequality \( y > x \) does not hold. Understanding the characteristics of each quadrant helps us to conclude accurately where the inequality conditions are valid or invalid.