How Can You Use a Number Line to Represent Solutions of an Inequality?

A number line visually represents the solutions of an inequality by shading the range of values that satisfy the inequality and using open or closed circles to indicate whether the endpoint is included.

Here's a breakdown of how to represent inequality solutions on a number line:

1. Draw a Number Line:

• Draw a straight line.
• Mark zero (0) at the center or a suitable point.
• Mark positive numbers to the right of zero and negative numbers to the left. Ensure consistent spacing between numbers.

2. Locate the Boundary Point:

• Identify the number involved in the inequality (e.g., x > 3, the boundary point is 3).
• Mark this number on the number line.

3. Use Open or Closed Circles:

• Open Circle (O): Use an open circle at the boundary point if the inequality is strict (i.e., uses only ">" or "<"). This indicates that the boundary point itself is not included in the solution. For example, x > 3 uses an open circle at 3.
• Closed Circle (●): Use a closed (filled-in) circle at the boundary point if the inequality is inclusive (i.e., uses "≥" or "≤"). This indicates that the boundary point is included in the solution. For example, x ≤ 5 uses a closed circle at 5.

4. Shade the Solution Region:

• For "greater than" (>) or "greater than or equal to" (≥): Shade the number line to the right of the boundary point. This indicates all numbers greater than the boundary point are solutions.
• For "less than" (<) or "less than or equal to" (≤): Shade the number line to the left of the boundary point. This indicates all numbers less than the boundary point are solutions.

Example:

Let's represent the inequality x > -2 on a number line:

1. Draw a number line.
2. Mark -2 on the number line.
3. Since the inequality is "x > -2" (greater than), use an open circle at -2.
4. Shade the number line to the right of -2.

This shaded region with the open circle at -2 represents all values of x that are greater than -2.

In summary, a number line uses a combination of open or closed circles at specific points and shading to represent the infinite number of possible solutions to inequalities.