Problem 3 - Algebra A Combined Function

Question

Graph each inequality on the number line. $$ x>\frac{1}{2} $$

Step-by-Step Solution

Answer

An open circle at \( \frac{1}{2} \), shaded to the right.

Step 1: Understand the inequality

The inequality is given as \( x > \frac{1}{2} \). This means that the values of \( x \) are all numbers greater than \( \frac{1}{2} \).

Step 2: Identify the type of circle

Since the inequality is \( x > \frac{1}{2} \), a hollow (or open) circle is used to represent that \( \frac{1}{2} \) is not included in the solution.

Step 3: Locate \( \frac{1}{2} \) on the number line

Place an open circle at \( \frac{1}{2} \) on the number line. This indicates that \( \frac{1}{2} \) itself is not part of the solution set.

Step 4: Shade the region representing the solution

Starting from the open circle at \( \frac{1}{2} \), shade the number line to the right. This shows all the numbers greater than \( \frac{1}{2} \), indicating the solution set toward positive infinity.

Key Concepts

Number Line

A number line is a straight horizontal line that helps visually represent numbers. It is marked at regular intervals to represent integers, fractions, and sometimes even decimals. In mathematics, number lines are a practical tool for understanding various concepts, including inequalities. On a number line, each point corresponds to a number, making it easier to see the relationships between different values. When working with inequalities, a number line can show which numbers are solutions and which are not. By plotting values on it, we can quickly determine where solutions fall and how to represent them effectively.

Open Circle

The concept of an open circle is essential when graphing inequalities on a number line. An open circle, often drawn as a hollow circle, represents a boundary that is not included in the solution set. In the inequality \( x > \frac{1}{2} \), the open circle is placed at \( \frac{1}{2} \) to indicate that this value is not part of the solutions. This open circle tells us not to include \( \frac{1}{2} \), as \( x \) should only be the numbers greater than this. Thus, it serves as a visual cue to show exclusion on the number line.

Shading

Shading is a crucial step in graphing inequalities since it visually demonstrates the part of the number line that contains all possible solutions. After drawing an open circle on the number line at the boundary \( \frac{1}{2} \), the next step is shading. For the inequality \( x > \frac{1}{2} \), shade the line to the right of the open circle. This shading direction points towards increasing values and includes all numbers greater than \( \frac{1}{2} \). Shading indicates the "greater than" aspect of the inequality clearly, helping anyone reading the graph to easily identify the solution range.

Inequality Solutions

Inequality solutions involve finding all the possible values that satisfy a given inequality. This requires not only identifying the numbers but also properly representing them on a number line. For instance, with \( x > \frac{1}{2} \), the solution set is all the numbers greater than \( \frac{1}{2} \). It's important to note that these solutions continue to positive infinity, showing a never-ending set of answers. By drawing an open circle at \( \frac{1}{2} \) and shading to the right, we properly capture the inequality's solutions on a number line. This effective visualization helps in understanding and solving inequality problems more efficiently.