Problem 21 - Thinking Mathematically

Question

Graph each linear inequality. \(y \geq 0\)

Step-by-Step Solution

Answer

The graph of the inequality \(y \geq 0\) is a shading of the region above the x-axis, including the x-axis.

Step 1: Identify the Boundary Line

The boundary line for this inequality is \(y = 0\). This is the x-axis itself. Since the inequality is \(y \geq 0\), the boundary line will be a solid line as it includes the points on the line \(y = 0\).

Step 2: Identify the Shaded Region

Since the inequality is \(y \geq 0\), the shaded area will be above the x-axis.

Step 3: Graph the Inequality

Draw a solid line representing the boundary line \(y = 0\), which is the x-axis. As the inequality is \(y \geq 0\), shade the region above the x-axis. This shaded region denotes the required solution for the inequality.

Key Concepts

Boundary Line in Inequalities

In graphing linear inequalities, the boundary line is a crucial element. Think of it as a 'fence' separating two different areas. This line itself is based on the related linear equation you get by replacing the inequality symbol with an equal sign. The main point to remember is that the type of line—dashed or solid—depends on the inequality symbol used.

For example, if your inequality symbol includes an 'or equal to' component, like \(y \geq 0\) or \(x \leq 5\), the boundary line is solid, indicating that points on the line are part of the solution. Conversely, if the inequality lacks the 'or equal to' part, such as \(y > 0\) or \(x < 2\), then the boundary line will be dashed, suggesting points on the line do not satisfy the inequality. It's essential to make this distinction clear on your graph to properly convey the solutions set of the inequality.

Shading Regions in Inequality Graphs

Once the boundary line is drawn, the next step in graphing linear inequalities is shading the correct region. The shaded area represents all possible solutions to the inequality. To figure out which side to shade, choose a test point not on the boundary line—often \(0,0\) if it's not on the line—and plug its values into the inequality.

If the inequality holds true, then the region containing the test point is your solution region. If it's false, you shade the opposite side. In our example with \(y \geq 0\), we shade above the x-axis because that area includes all the points where the y-values are greater than or equal to zero. Remember: The shading is what provides a visual representation of all the solutions that satisfy the inequality, making it an integral part of the graphing process.

Inequality Symbols and Meanings

Understanding the symbols used in inequalities is key to representing them correctly in graph form. Each inequality symbol indicates a different range of possible values that satisfy the inequality. Here's a breakdown:

• \(< >\) - 'Greater than': Y-values are exclusively greater than a certain number, but do not include the number itself.
• \(\geq\) - 'Greater than or equal to': Y-values are greater than a certain number, including the number itself.
• \(< <\) - 'Less than': Y-values are exclusively less than a certain number, again without including the number itself.
• \(\leq\) - 'Less than or equal to': Y-values are less than a certain number and include the number itself.

These symbols define whether or not the boundary line is part of the solution set and determine the direction in which we shade the graph. Gaining familiarity with the meanings behind these symbols will significantly aid in interpreting and solving inequality problems.