Problem 28 - Prealgebra

Question

Graph each inequality on a number line. $$n<11$$

Step-by-Step Solution

Answer

Shade the number line to the left of an open circle at 11.

Step 1: Understand the Inequality

The inequality given is \( n < 11 \). This means that the values of \( n \) are all numbers that are less than 11.

Step 2: Identify the Boundary Point

The boundary point is the number that \( n \) is compared to, which is 11 in this case. The inequality is strict (less than, not less than or equal to), so 11 is not included in the solution set.

Step 3: Draw the Number Line

Draw a horizontal line and mark a point for several numbers around 11 on this line, including 11 itself for clarity.

Step 4: Plot the Boundary Point

On the number line, place an open circle (not filled in) at 11. This indicates that 11 is not part of the solution.

Step 5: Shade the Solution Region

Shade the portion of the number line to the left of the open circle at 11 to show all the numbers that are less than 11. This shaded region represents the solution to the inequality \( n < 11 \).

Key Concepts

Understanding the Number Line

The number line is a simple graphical representation that helps visualize numbers and their relationships. In this exercise, we use a horizontal line to plot numbers.

• Start by drawing a straight, horizontal line.
• Evenly space the numbers around the boundary point, in our case around 11, to provide context.
• Mark essential points to have meaningful visualization for any inequality or equation.

When graphing inequalities, the number line serves as a tool to show which numbers meet the conditions set by the inequality.

Discovering the Boundary Point

A boundary point is crucial in solving inequalities. It is the number the variable is compared to in an inequality.
In our example, the boundary point is 11:

• This is because the variable, denoted as \( n \), is being compared to 11 in the inequality \( n < 11 \).
• Whether or not the boundary point itself is part of the solution depends on the type of inequality.

Since the inequality sign is less than (not less than or equal to), 11 is not included. Hence, we represent it with an open circle when plotted on the number line.

Finding the Inequality Solution

The inequality solution consists of all numbers that satisfy the given inequality.
For \( n < 11 \):

• Any number less than 11 meets the criteria of \( n < 11 \).
• To graphically display these numbers, move to the left of 11 on the number line.

This clear representation allows us to easily identify which numbers form part of our solution by checking the shaded section of the number line.

Shading the Solution Region

Shading the solution region is the final step in graphing an inequality on a number line. It visually represents all potential solutions.
Here's how to shade the region correctly for \( n < 11 \):

• Start from the boundary point, which is clearly marked by an open circle at 11.
• Proceed to shade everything extending to the left of 11, showing all numbers less than 11.

This shaded area is our solution set, and visually it communicates the range of numbers that satisfy the inequality condition. By looking at it, anyone can tell which numbers are included and excluded in the solution.