91Ó°ÊÓ

In graphing linear inequalities, understanding the slope-intercept form is essential. This form is expressed as \( y = mx + b \), where \( m \) stands for the slope, and \( b \) is the y-intercept. The slope determines the steepness and direction of the line, while the y-intercept identifies the point where the line crosses the y-axis.

In our exercise, the inequality \( y > \frac{1}{3}x + 5 \) follows this form. Specifically, the slope \( m \) is \( \frac{1}{3} \), indicating that for every 3 units moved horizontally, the line rises by 1 unit vertically. The y-intercept \( b \) is 5, meaning the graph will start at the point (0, 5) on the y-axis. This helps establish the initial framework required for constructing the line on a graph.

When graphing the inequality \( y > \frac{1}{3}x + 5 \), the boundary line serves as the foundation. It's derived from the inequality by treating it as an equation: \( y = \frac{1}{3}x + 5 \).

To plot this line, start at the y-intercept, (0, 5), on the graph. Using the slope of \( \frac{1}{3} \), move up 1 unit and to the right 3 units to obtain another point at (3, 6). Connecting these points forms the boundary line. However, since the inequality is \( y > \), the boundary line will be dashed. This dashed line indicates that points on the line are not included in the solution set of the inequality, clarifying its exclusivity.

Shading regions is a crucial part of representing inequalities on graphs. It visually indicates the solution set. For \( y > \frac{1}{3}x + 5 \), shading the correct region is important because it shows all the possible \( y \)-values that satisfy the inequality.

In this case, since our inequality is \( y > \), you would shade the region above the dashed boundary line. This shaded area reflects where the y-values are greater than those on the line itself. To ensure you've shaded the correct region, pick a test point that's not on the line, such as (0, 0). Substitute these values into the inequality: \( 0 > \frac{1}{3}(0) + 5 \), which is not true. Thus, the opposite region (above the line) is the correct shaded area indicating all solutions to the inequality.