Problem 7 - Prealgebra

Question

Graph each inequality on a number line. $$y \leq 14$$

Step-by-Step Solution

Answer

Shade the number line to the left of 14 and include a closed circle at 14.

Step 1: Understanding the Inequality

The inequality provided is \( y \leq 14 \). This means that the variable \( y \) includes all values that are less than or equal to 14.

Step 2: Identifying Boundary Point

The boundary point for the given inequality is 14. It is included in the solution set because the inequality is 'less than or equal to'.

Step 3: Marking the Boundary Point

On the number line, place a closed circle (or filled dot) at 14 to indicate that this point is included in the solution set.

Step 4: Shading the Solution Region

Shade the number line to the left of the closed circle at 14. This indicates that all values less than 14, as well as 14 itself, satisfy the inequality.

Key Concepts

Inequalities on a Number Line

When dealing with inequalities, like the one given as \( y \leq 14 \), a number line is a powerful visual tool. A number line helps to clearly visualize the range of values that satisfy the inequality. In the context of \( y \leq 14 \), you need to show all numbers starting from negative infinity up to 14.
This includes all the numbers that are less than 14, and 14 itself. A number line is particularly useful because:

• It visually demonstrates the continuous nature of solutions for inequalities.
• It helps differentiate between strict inequalities (like \( y < 14 \)) and inclusive inequalities (like \( y \leq 14 \)).

Understanding how to graph inequalities on a number line is crucial for solving and interpreting mathematical problems accurately.

Closed Circle on a Number Line

To indicate that the boundary value is included in the solution set for an inequality, we use a closed circle on a number line. For the inequality \( y \leq 14 \), 14 is a part of the solution set because the inequality sign \( \leq \) means "less than or equal to".
Thus, you represent this point with a closed circle (or a filled dot) above the number 14 on your number line.
The closed circle tells you:

• The boundary point 14 is part of the solution set.
• The solution includes every number less than or equal to the boundary point.

This concept helps differentiate between included and excluded boundary values in graphical representations of inequalities.

Shading Solution Region

Shading the solution region on a number line indicates which values satisfy the inequality. For \( y \leq 14 \), you place the closed circle at 14, and then shade all numbers to the left of this circle.
Shading shows:

• Every number in the shaded region is a solution to the inequality.
• It extends infinitely to the left, encompassing all lesser values below 14.

This method of shading is important because it provides a clear and comprehensive visualization of all the possible solutions to given inequality.
When combined with a closed circle, it ensures that you visually communicate the inclusion or exclusion of specific boundary points.