Problem 7 - A Graphical Approach to College Algebra

Question

Graph each inequality. Do not use a calculator. $$3 x-5 y>6$$

Step-by-Step Solution

Answer

Graph the line \(3x - 5y = 6\) with a dashed line and shade the region opposite to (0,0).

Step 1: Rewrite the Inequality as an Equation

Start by rewriting the given inequality as an equation by replacing the inequality sign with an equal sign.\[3x - 5y = 6\]

Step 2: Find the Intercepts

To find the x-intercept, set \(y = 0\) in the equation \(3x - 5y = 6\):\[3x = 6 \implies x = 2\] So the x-intercept is at \((2, 0)\).To find the y-intercept, set \(x = 0\):\[-5y = 6 \implies y = -\frac{6}{5}\]So the y-intercept is at \((0, -\frac{6}{5})\).

Step 3: Plot the Intercepts and Graph the Line

Plot the x-intercept (2, 0) and the y-intercept (0, -\(\frac{6}{5}\)) on the coordinate plane. Draw a dashed line through these points to represent the equation \(3x - 5y = 6\). Use a dashed line because the original inequality is not inclusive (\(>\), not \(\geq\)).

Step 4: Determine Which Side Satisfies the Inequality

Choose a test point not on the line, such as (0,0), to determine which side of the line to shade. Substitute (0,0) into the inequality:\[3(0) - 5(0) > 6 \implies 0 > 6\]This is false, meaning the region not containing (0,0) satisfies the inequality. Shade the side of the line opposite to the origin.

Key Concepts

Coordinate Plane

The coordinate plane is a crucial tool for graphing inequalities in math. It consists of two perpendicular lines called the x-axis and y-axis, which divide the plane into four quadrants. Every point on the plane is defined by its coordinates, written in the form \(x, y\). The origin, where both axes meet, is indicated by the coordinates \(0, 0\).

When graphing inequalities, the coordinate plane allows you to visualize the relationships between variables. It provides a clear way to see where the solutions of an inequality lie. For each point on the plane, the x-coordinate tells you how far left or right the point is from the origin, while the y-coordinate tells you how far up or down the point is. This visualization helps identify parts of the plane where an inequality holds true.

X-Intercept

The x-intercept is where a graph crosses the x-axis. At this point, the y-value is always zero.

Finding the x-intercept involves setting the y-variable in the equation to zero and solving for x. This helps you understand where the graph touches or crosses the x-axis. For instance, in the equation from the exercise, \(3x - 5y = 6\), setting \(y = 0\) gives us \(3x = 6\), resulting in the x-intercept \((2, 0)\).

This point is vital for sketching the graph as it provides one of the two anchor points needed to draw the line. The x-intercept always gives you a reliable point to start plotting on the coordinate plane.

Y-Intercept

The y-intercept is the point where the graph crosses the y-axis. Here, the x-value is zero since it is on the y-axis.

To find the y-intercept, we set the x-variable in the equation to zero and solve for y. This reveals the point at which the graph touches the y-axis. Using the exercise equation \(3x - 5y = 6\), when \(x = 0\), solving gives \(-5y = 6\), or \(y = -\frac{6}{5}\). Thus, the y-intercept is at \((0, -\frac{6}{5})\).

This intercept provides the second anchor point for drawing the line on the coordinate plane. Together with the x-intercept, it forms the basis for graphing the line accurately.

Dashed Line

A dashed line is used in graphing inequalities when the inequality sign is 'greater than' (>) or 'less than' (<), because the boundary line is not included in the solution set.

For an inequality like \(3x - 5y > 6\), we first convert it to an equation to find the boundary line, \(3x - 5y = 6\). However, because the inequality does not include equal to (\(\ge\) or \(\le\)), the line itself is not part of the solution. Thus, we use a dashed line to indicate that points on the line are not solutions to the inequality.

Drawing a dashed line differentiates clearly between the areas where the inequality holds and where it does not. It enhances the clarity of the graph, showing students which part of the plane they need to focus on for solutions.