Problem 24 - Elementary Algebra Within Reach

Question

Solve and graph the inequality. $$z-2>0$$

Step-by-Step Solution

Answer

The solution to the inequality \( z-2 > 0 \) is \( z > 2 \), which is represented on a number line with an open circle at 2 and a ray extending to the right.

Step 1: Solve the Inequality

To solve the inequality, isolate \( z \) by adding 2 to both sides of the inequality: \( z-2+2 > 0+2 \). This simplifies to \( z > 2 \). This means that \( z \) can be any number greater than 2.

Step 2: Graph the Solution

The graph is a number line that represents all possible values of \( z \). Since \( z \) is greater than 2, start with an open circle at 2. Then, draw a ray going to the right from 2, indicating that \( z \) contains all numbers greater than 2. An open circle is used because 2 is not included in the solution.

Key Concepts

Graphing Inequalities

Understanding the concept of graphing inequalities is essential for visualizing solutions on a number line. Inequalities express a range of values that satisfy a given condition, rather than pinpointing a specific value. For example, in the inequality \( z > 2 \), we are looking for all values of \( z \) that are greater than 2.

When graphing this inequality, the focus is on highlighting the portion of the number line that represents these valid solutions. Here's how you can approach it:

• Identify the boundary point—here it's the number 2.
• Determine whether the boundary point itself is part of the solution set. In \( z > 2 \), the number 2 is not included.
• Illustrate by shading or extending the graph in the direction where the inequality holds true.

This approach gives a clear visual understanding of where the solution to the inequality lies on a number line.

Number Line Representation

A number line is a powerful tool for representing the solutions to inequalities. It allows us to place numbers in a simple, clear layout, which can make it easier to understand abstract mathematical concepts. In the case of \( z > 2 \), the number line will:

• Start with a point on the line labeled as 2, representing the boundary condition.
• Use an open circle at 2 to indicate that this number is not included in the solution.
• Have a ray or shaded segment extending to the right, showing that all numbers greater than 2 satisfy the inequality.

Number lines provide a straightforward visual aid by mapping mathematical ideas into a format that is easy to interpret. They also help reinforce understanding by providing a real-world representation of an inequality's solution set.

Open Circle in Graphing

When graphing inequalities like \( z > 2 \) on a number line, the open circle plays a critical role. It is used to denote that a particular value, in this case, the number 2, is not part of the solution set. An open circle is placed directly on the number line at the boundary point.

Here's what the open circle communicates about your inequality:

• An open circle means that the endpoint is excluded from the solution.
• This is often the case with strict inequalities, such as \( > \) or \( < \), where the solution includes numbers on only one side of a boundary.

In contrast, a closed circle would indicate the inclusion of the boundary number in the solution set (used for \( \geq \) or \( \leq \)). Understanding this distinction between open and closed circles is crucial for accurately interpreting and graphing inequalities.