Problem 15 - Algebra 2

Question

Graph each inequality. $$ y > \frac{1}{3} x+5 $$

Step-by-Step Solution

Answer

Graph the line \( y = \frac{1}{3}x + 5 \) with a dashed line and shade above it.

Step 1: Identify Slope and Y-intercept

The inequality given is in the form of a linear equation, which can be written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. From the inequality, we identify that the slope \( m \) is \( \frac{1}{3} \) and the y-intercept \( b \) is 5.

Step 2: Graph the Boundary Line

To graph the inequality \( y > \frac{1}{3}x + 5 \), begin by graphing the boundary line \( y = \frac{1}{3}x + 5 \). Start at the y-intercept (0, 5) and use the slope to find other points. With a slope of \( \frac{1}{3} \), move up 1 unit and right 3 units to plot another point at (3, 6). Connect these points with a dashed line, indicating the inequality is not inclusive (\( > \)).

Step 3: Shade the Appropriate Region

Since the inequality is \( y > \frac{1}{3}x + 5 \), shade the region above the dashed line. This area represents where the y-values are larger than those on the line.

Key Concepts

Slope-Intercept Form

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.

Boundary Line

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

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.