Problem 40 - Intermediate Algebra

Question

Graph each inequality. $$ x<\frac{1}{6} y $$

Step-by-Step Solution

Answer

Graph the inequality by drawing a dashed line for \(x=\frac{1}{6}y\) and shade the region to the right of the line.

Step 1: Understand the Equation

The given inequality is \(x < \frac{1}{6}y\). This inequality can be treated as a linear inequality. Our goal is to graph this inequality on a coordinate plane.

Step 2: Convert Inequality to Equation

To find the boundary of the inequality, first, convert the inequality into an equation: \(x = \frac{1}{6}y\). This will help us to identify the line that serves as the boundary for the inequality.

Step 3: Draw the Boundary Line

The equation \(x = \frac{1}{6}y\) can be rewritten as \(y = 6x\) for easier graphing. This is a straight line, which passes through the origin (0,0) and has a slope of 6. Since the original inequality is strict (\(x < \frac{1}{6}y\)), draw a dashed line to indicate that the points on the line are not included in the solution.

Step 4: Choose a Test Point

To determine which side of the boundary line represents the solution to the inequality, choose a test point not on the line. A simple choice is the point (0,0). Substitute this point into the original inequality: \(0 < \frac{1}{6}(0)\). This simplifies to \(0 < 0\), which is false.

Step 5: Identify the Solution Region

Since the test point (0,0) does not satisfy the inequality, the solution region is the side of the line that does not include the origin (0,0). Shade the region on the graph that is to the right of the line \(y = 6x\), as this represents \(x < \frac{1}{6}y\).

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can graph points, lines, and curves to represent mathematical equations and inequalities visually. It consists of two axes: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at a point called the origin, denoted as (0, 0). The coordinate plane divides into four regions, known as quadrants, which help locate and reference points on the plane.
The importance of the coordinate plane in graphing is that it allows us to visualize mathematical relationships and understand how variables interact. When working with inequalities like \(x < \frac{1}{6}y\), the coordinate plane helps show which parts of the space meet the conditions set by the inequality.

Boundary Line

A boundary line is a straight line that separates the solutions of an inequality into different regions on a graph. For the inequality \(x < \frac{1}{6}y\), you first convert it to the equation \(x = \frac{1}{6}y\) to find this boundary line. By doing so, it becomes easier to see where the solutions lie relative to the line.
Boundary lines can be solid or dashed. If the inequality is non-strict (less than or equal, or greater than or equal), like \(x \leq \frac{1}{6}y\), the line is solid, because the points on the line are part of the solution set. For strict inequalities, like \(x < \frac{1}{6}y\), the line is dashed to show that the points on the line are not included in the solution.

Slope

The slope of a line describes its steepness and direction, which is crucial for understanding and graphing linear equations. In general, the slope is calculated as the change in y over the change in x, denoted as \(m = \frac{\Delta y}{\Delta x}\).
In the equation \(y = 6x\), which represents the boundary line corresponding to \(x = \frac{1}{6}y\), the slope is 6. This indicates that for every 1 unit increase in the x-direction, the y-value increases by 6 units. Slopes can be positive, negative, zero, or undefined, and dictate the direction and angle of the line on the coordinate plane.

Inequality Graphing Techniques

When graphing inequalities, it is essential to determine the boundary line and which side of this line contains the solutions to the inequality. Here are some techniques to keep in mind:

• Convert the inequality to an equation first to identify the boundary line.
• Determine the type of boundary line: use a solid line for \(\leq\) or \(\geq\), and a dashed line for \(<\) or \(>\).
• Select a test point not on the line (commonly the origin, unless the line passes through it) to see which side of the line satisfies the inequality.
• Shade the region where the inequality holds true, indicating the solution set.

By following these steps, you can graph any linear inequality, like \(x < \frac{1}{6}y\), effectively and correctly on a coordinate plane.