91Ó°ÊÓ

In graphing inequalities, the boundary line is crucial as it helps delineate which area represents the solutions to the inequality. When dealing with an inequality like \(3x \leq 18\), the first step is to solve for \(x\). By dividing both sides by 3, we isolate \(x\) and find that the inequality becomes \(x \leq 6\). This means the boundary line for the inequality's graphical representation is \(x = 6\).

When graphing the boundary line on a coordinate plane, it's essential to understand whether the line is solid or dashed. For \(x \leq 6\), since the inequality includes \(x = 6\) as part of the solution, the boundary line will be solid. If it were a strict inequality, like \(x < 6\), the line would be dashed, indicating that points directly on it are not included.

Graphing an inequality requires representing it visually on a coordinate plane, which is a two-dimensional surface defined by a vertical y-axis and a horizontal x-axis. Each point on this plane is defined by a pair of coordinates \((x, y)\). To visualize the inequality \(x \leq 6\), you draw this on the plane. Here, our interest lies in a vertical line where \(x = 6\), with every point on that line having an x-coordinate of 6.

While graphing, placing the boundary line accurately along this vertical path is essential. Make sure to label your axes and plot accurately to reflect the inequality's conditions. Transforming algebraic inequalities to graphical representations gives clarity on the solutions it describes.

The solution to an inequality is represented as the set of values that satisfy the inequality condition. In our example, \(x \leq 6\), we look for all \(x\) values that are smaller than or equal to 6. These values become the solution set. Testing specific points can help verify which side of the boundary line holds true for the inequality:

• For \(x < 6\), choose a value like 5. Testing gives \(5 \leq 6\), which is true. Thus, this region is part of the solution.
• For \(x > 6\), choose a value like 7. Testing gives \(7 \leq 6\), which is false. This region is not included in the solution.

Testing points clarifies which side of the boundary line encompasses the solutions, aiding in the shading process on the graph.

The shaded region on a graph visually indicates the set of solutions to an inequality. With \(3x \leq 18\), after determining that \(x \leq 6\), you know to shade the region where \(x\) is less than or equal to 6.To graph this, once the boundary line \(x = 6\) is plotted, shading is added to the left of this line. The shading must clearly convey that every point in this region satisfies the inequality.

Make sure the shading is distinct and covers all necessary areas towards negative infinity on the x-axis. This helps anyone viewing the graph to immediately recognize the solution region and understand the inequality's coverage.