How to Graph Inequalities on a Number Line

Learning to graph inequalities on a number line is a key skill in algebra. This skill helps us see and solve problems involving numbers. Inequalities show how one number compares to another. We use symbols like:

• Greater than (>)
• Less than (<)
• Greater than or equal to (≥)
• Less than or equal to (≤)

Let’s go through the steps to graph inequalities step by step.

What Are Inequalities?

Inequalities are statements that show a relationship between two values. They tell us whether one value is less than, greater than, less than or equal to, or greater than or equal to another value.

For example, if we say ( x > 3 ), it means that ( x ) can be any number that is greater than 3.

Steps to Graph Inequalities

1. Identify the Inequality
   First, figure out what type of inequality you have. Is it >, <, ≥, or ≤? This will tell you how to show the solution on the number line.

2. Find Key Points
   Find the number on the number line that matches your inequality. For ( x > 3 ), the key point is 3. For ( x \geq 3 ), the key point is still 3, but it’s a little different.

3. Choose Open or Closed Circles
   Now, we need to decide how to mark the key point:

   • Use an open circle for > or <. An open circle means the number is not included in your solution. For ( x > 3 ), put an open circle at 3.

   • Use a closed circle for ≥ or ≤. A closed circle means the number is included in the solution. For ( x \geq 3 ), put a closed circle at 3.

4. Shade the Right Area
   After placing your circle, the next step is to shade the number line to show all possible solutions:

   • For ( x > 3 ), shade all numbers to the right of 3, showing that any number larger than 3 is part of the solution.

   • For ( x < 3 ), shade to the left of 3.

   • For ( x \geq 3 ), shade to the right of the closed circle at 3, including 3 itself.

   • For ( x \leq 3 ), shade to the left of the closed circle at 3, including 3.

5. Examples
   Let’s look at a couple of examples to make it clear:

   • For ( x < 4 ):
     • Place an open circle at 4.
     • Shade to the left, meaning all values less than 4 are included.
   • For ( x \geq -1 ):
     • Place a closed circle at -1.
     • Shade to the right, meaning all values greater than or equal to -1 are included.

6. Compound Inequalities
   Sometimes, you may see compound inequalities like ( 2 < x \leq 5 ). For this:

   • Place an open circle at 2 and shade to the right.
   • Place a closed circle at 5 and shade to the left, stopping at the closed circle.

7. Multiple Inequalities
   If you have more than one inequality that affects the same variable, look for the overlap. For example, with ( x > 2 ) and ( x < 4 ), graph both and find the shaded area that overlaps (between 2 and 4).

8. Real-Life Uses
   Graphing inequalities can help us with real-life problems. For example, if you’re creating a budget, knowing that your expenses ( x ) need to be below a certain amount can be shown with an inequality.

9. Boundaries
   Think of inequalities as boundaries. The number line shows where these limits are.

10. Tips for Success

• Always check if you’re using an open or closed circle.
• Remember, the circle shows if a number is in the solution or not.
• When shading, make sure to extend your shading far enough to show all possibilities.

By following these steps, you'll understand how to graph inequalities on a number line. This skill will help you in school and give you a strong base for more advanced math topics.

Practice Makes Perfect!
The more you practice graphing inequalities, the better you will get. Start with simple inequalities and then try more complex ones. Soon, you will feel comfortable working with the number line, which will help you a lot in your studies!