Basics Of Graphing Inequalities

🔑 Key Concepts

• An inequality compares two quantities using symbols such as \( < \), \( > \), \( \le \), or \( \ge \).
• A closed (filled) circle means the number is included in the solution.
• An open circle means the number is not included in the solution.
• The arrow shows all numbers that make the inequality true.
• Numbers to the right are greater, and numbers to the left are smaller on a number line.

✏️ Worked Examples — Graphing Inequalities on a Number Line

Example 1

What this means:
The variable \(x\) can equal \(1\) or be any number greater than \(1\).

Why is the circle shaded?
The filled circle shows that \(1\) is included in the solution.

Direction of the arrow:
The arrow points right because numbers greater than \(1\) are located to the right.

Examples that work:
\(1, 2, 5, 10\)

Examples that do not work:
\(0, -3, -10\)

Example 2

What this means:
The variable \(x\) can be any number less than \(1\), but it cannot equal \(1\).

Why is the circle open?
The open circle shows that \(1\) is not included in the solution.

Direction of the arrow:
The arrow points left because numbers less than \(1\) are located to the left.

Examples that work:
\(0, -1, -4, -10\)

Examples that do not work:
\(1, 2, 5\)

Example 3

What this means:
The variable \(x\) can be any number greater than \(-1\), but it cannot equal \(-1\).

Why is the circle open?
The open circle shows that \(-1\) is not included in the solution.

Direction of the arrow:
The arrow points right because numbers greater than \(-1\) are located to the right.

Examples that work:
\(0, 2, 10, 100\)

Examples that do not work:
\(-1, -2, -5\)

🧠 Math Vocabulary

• Inequality: a mathematical statement comparing two values.
• Solution: a value that makes the inequality true.
• Closed Circle: indicates the endpoint is included.
• Open Circle: indicates the endpoint is not included.
• Number Line: a visual representation of numbers in order.

💡 Main Idea

Graphing inequalities means showing all numbers that make an inequality true on a number line. A closed circle is used when the number is included in the solution, such as with \( \le \) or \( \ge \). An open circle is used when the number is not included, such as with \( < \) or \( > \). The direction of the arrow shows whether the solution includes numbers greater than or less than the given value.

📚 What You Should Already Know

Students should know how to read and interpret a number line, understand positive and negative numbers, and compare numbers using symbols such as greater than and less than.

🚀 What Comes Next

After learning to graph simple inequalities, students will move on to solving one-step and multi-step inequalities, writing inequalities from word problems, and graphing compound inequalities.

🧩 Embed This Video in Your LMS

Teachers can embed this video directly into Canvas, Google Classroom, or another learning platform. Click below to copy the embed code.