Solve and Graph Inequalities — 3 Examples

🔑 Key Concepts

  • Solving an inequality uses many of the same steps as solving an equation.
  • When you divide or multiply both sides by a negative number, the inequality symbol must reverse.
  • A closed circle means the endpoint is included in the solution.
  • An open circle means the endpoint is not included in the solution.
  • Graphing the solution on a number line helps show all values that make the inequality true.

✏️ Worked Examples

🧠 Math Vocabulary

  • Inequality: A math sentence that compares two quantities using symbols such as \( < \), \( > \), \( \le \), or \( \ge \).
  • Solution set: All values that make an inequality true.
  • Closed circle: A point on the number line showing that the endpoint is included in the solution.
  • Open circle: A point on the number line showing that the endpoint is not included in the solution.
  • Reverse the inequality: When multiplying or dividing by a negative number, the inequality symbol must switch directions.

💡 Main Idea

Solving inequalities is very similar to solving equations, but there is one major difference: if you multiply or divide both sides by a negative number, the inequality symbol must reverse. After solving, graph the solution on a number line using a closed circle for included values and an open circle for values that are not included. The arrow shows all the numbers that continue to make the inequality true.

📚 What You Should Already Know

Before working through this lesson, students should be comfortable combining like terms, using inverse operations to isolate a variable, working with integer rules, and reading points on a number line. It also helps to understand the difference between equations and inequalities before graphing solution sets.

🚀 What Comes Next

After mastering one-step and multi-step inequalities, students are ready for compound inequalities, inequality word problems, and graphing more complex solution sets. This skill also helps prepare students for systems of inequalities and later algebra topics where understanding solution regions becomes even more important.

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