Problem 69 - A Graphical Approach to College Algebra

Question

Use the shading capabilities of your graphing calculator to graph each inequality or system of inequalities. $$3 x+2 y \geq 6$$

Step-by-Step Solution

Answer

Shade the region above the line \(3x + 2y = 6\).

Step 1: Rewrite the Inequality as an Equation

First, to graph the inequality, we begin by rewriting it as an equation. The given inequality is \(3x + 2y \geq 6\). Rewrite it as an equation: \(3x + 2y = 6\). This will help us find the boundary line.

Step 2: Find the Boundary Line Intercepts

To graph the boundary line, find the intercepts. For the x-intercept, set \(y = 0\) and solve for \(x\): \(3x = 6\), giving \(x = 2\). For the y-intercept, set \(x = 0\) and solve for \(y\): \(2y = 6\), giving \(y = 3\). This means the intercepts are (2, 0) and (0, 3).

Step 3: Plot the Boundary Line on the Graph

Using the intercepts (2, 0) and (0, 3), draw the line \(3x + 2y = 6\) on the graph. Since the inequality is \(\geq\), the line should be solid, indicating that points on the line satisfy the inequality.

Step 4: Determine the Shaded Region

Choose a test point not on the line to determine which side to shade. The origin \((0, 0)\) is a convenient test point. Substitute \((0, 0)\) into the inequality \(3x + 2y \geq 6\): \(0 \geq 6\), which is false. Therefore, shade the opposite side from the origin, which means the region that includes the line above the x-axis.

Step 5: Use Your Graphing Calculator's Shading Function

Input the inequality \(3x + 2y \geq 6\) into your graphing calculator. Use its shading feature to automatically shade the region where the inequality holds. The calculator should shade the region above the line \(3x + 2y = 6\).

Key Concepts

graphing calculator

Using a graphing calculator can significantly ease the process of graphing inequalities, especially when dealing with complex systems or multiple equations. These calculators are equipped with functions that allow you to visualize inequalities in a clear, precise manner.

To begin, you need to input the inequality into the calculator. Many graphing calculators have a specific menu or option for inequalities, where you can simply enter the expression. Make sure to use the correct symbols, such as:

• ">=" or "≤" for greater than or equal to and less than or equal to.
• ">" or "<" for greater than and less than.

The graphing calculator will then automatically compute and display the solution set by shading the appropriate region of the graph. This shading visually represents where the inequality holds true, thus aiding in understanding which parts of the graph satisfy the inequality.

boundary line

A boundary line plays a crucial role in graphing inequalities as it separates the coordinate plane into different regions. In the case of the inequality given, the boundary line is derived from the related equation obtained by replacing the inequality symbol with an equal sign.

For instance, in the inequality \(3x + 2y \geq 6\), the boundary line equation is \(3x + 2y = 6\). This line is graphically represented by using its intercepts, plotting the points where it crosses the x-axis and y-axis.

It's important to note if the inequality includes "equal to" (\(≥\) or \(≤\)), the boundary line will be drawn solid to indicate that the points on this line are part of the solution set. If the inequality was strict (\(>\) or \(<\)), you would draw a dashed line, signifying that the points on the line are not included in the solution.

shading regions

Shading regions in a graph of an inequality is the visual method of showing where the inequality holds true. Once the boundary line is plotted, you must determine which side of the line to shade.

To decide this, select a test point that is not on the boundary line. A common choice is the origin, \(0,0\), unless it lies on the line itself.

• Substitute this test point into the original inequality.
• If the inequality holds true, shade the region that includes the test point.
• If false, shade the opposite region.

For our inequality \(3x + 2y \geq 6\), testing the point \(0, 0\) gives \(0 \geq 6\), which is false. Therefore, you shade the region on the opposite side of the boundary line. This shading helps to create a clear visual distinction of the solution set on the graph.

x-intercept and y-intercept

Finding the x-intercept and y-intercept is an important step in graphing linear inequalities because these intercepts give you exact points to start drawing your boundary line.

• To find the x-intercept, set \(y = 0\) and solve for \(x\). For the equation \(3x + 2y = 6\), setting \(y = 0\) yields \(3x = 6\), resulting in \(x = 2\).
• To find the y-intercept, set \(x = 0\) and solve for \(y\). Using the same equation, setting \(x = 0\) gives \(2y = 6\), meaning \(y = 3\).

These intercepts, \((2, 0)\) and \((0, 3)\), are essential as they help you draw the boundary line correctly on the graph. With these points accurately plotted, your boundary line will be precise, ensuring your shaded region remains accurate and reliable for interpreting the solution to the inequality.