91Ó°ÊÓ

Understanding how to plot inequalities is a foundational skill in algebra and precalculus. It's essential to know that inequalities, much like equations, can be represented visually on a graph. However, unlike equations where you plot a single line, an inequality indicates a region.

To plot an inequality such as \( y > -7 \), start by identifying the related equation, which in this case is \( y = -7 \). This equation forms a boundary line on a graph. Because we're dealing with an inequality, not an equation, this boundary is represented by a dashed or dotted line rather than a solid one, indicating that the points on the line are not part of the solution set.

Why a dotted line? It conveys that points exactly on the line \( y = -7 \) are not included in the inequality \( y > -7 \). It's a visual cue telling us that, while the line \( y = -7 \) is a reference, the actual solutions lie in the area that does not touch this line.

After plotting the boundary line for an inequality, we employ the test point method to determine which side of the line should be shaded. This technique involves choosing a specific point that is not on the boundary line and substituting its coordinates into the inequality. A good default choice for a test point is usually the origin, \( (0,0) \), because of its simplicity, unless it lies on the boundary line itself.

In the example \( y > -7 \), we can take \( (0,0) \) as our test point. We substitute \(y\) with 0 to see if the inequality holds: \(0 > -7\), which is true. This tells us the origin is part of the solution set, thus the region that contains the origin – above the line – is the correct area to shade.

Why Not Always Use (0,0)?

The origin can't be used as a test point if the boundary line passes through it. In such a case, you'd select another point, ensuring it's easy to work with, like \( (1,0) \) or \( (0,1) \).

Once the boundary line is plotted and a test point is chosen, the next step is shading the graph. Inequality shading is a visual representation that helps us see all the solutions to an inequality on a graph.

If the inequality is '>' or '\( > \)', you shade above the line; if it's '<' or '\( < \)', shade below the line. When the inequality includes '=,' such as '\( \geq \)' or '\( \leq \)', the boundary line becomes solid, and shading indicates on which side the solutions reside, including points on the line itself.

For our example with \( y > -7 \), the appropriate area to shade is above the dotted line. This area represents all points where the \(y\)-value is greater than -7. Shading ensures the person looking at the graph can immediately identify the solution set, providing a clear visual distinction between solutions and non-solutions to the inequality.