Problem 26 - Prealgebra

Question

Graph each inequality on a number line. $$d \leq 5$$

Step-by-Step Solution

Answer

Plot a solid circle at 5, shade to the left on the number line.

Step 1: Understanding the Inequality

The inequality given is \( d \leq 5 \). This means that the value of \( d \) can be any number less than or equal to 5.

Step 2: Identify Boundary Point

The critical point here is 5. Since \( d \) can be equal to 5, we will represent 5 with a solid circle or dot on the number line.

Step 3: Represent the Direction of Inequality

Since \( d \) is less than or equal to 5, shade or draw a line to the left from the point 5 on the number line. This shows all numbers less than 5 are included in the solution set.

Step 4: Draw the Number Line

Draw a horizontal line and mark points on it, including 5 and surrounding numbers like 4 and 6 for context.

Step 5: Place the Solid Circle and Shade

Place a solid circle at 5 on the number line. Then shade or draw a line extending left from 5 to show that all numbers less than 5 are included.

Key Concepts

Number Line

A number line is a visual representation of numbers on a straight, horizontal line. It's useful for showing the position of numbers relative to each other. On a number line:

• Numbers increase in value as you move to the right.
• Numbers decrease in value as you move to the left.

To graph inequalities like \(d \leq 5\), number lines help us visualize all the possible solutions. First, draw the line and choose a range of numbers to display. You place a critical point, in this case, 5, and show directionality. In this exercise, your number line should clearly indicate 5 with a solid circle to show inclusion and then shade to the left, representing all numbers less than 5.

Inequalities

Inequalities are mathematical statements used to compare two values. They show relationships like less than, greater than, less than or equal to, and greater than or equal to. An inequality like \(d \leq 5\) suggests several values can satisfy it. In this case:

• \(d\) can be any value less than 5 such as 4, 3, or 0.
• \(d\) can also be equal to 5 due to the 'equal to' portion of \(\leq\).

Understanding inequalities helps solve problems that have more than one answer. It's important to recognize which part of the inequality relates to 'less than' and which part includes the boundary point, typically indicated with a solid or open circle on a number line.

Mathematical Representation

Mathematical representation is a way of visually displaying mathematical concepts, making them easier to understand. For inequalities, this often involves drawing number lines to show solution sets. When representing \(d \leq 5\) mathematically:

• Draw a number line with relevant points around the boundary, for instance, 4, 5, and 6.
• Mark the number 5 with a solid circle to show that 5 is included in the solutions.
• Shade the section of the line to the left of 5 to demonstrate numbers that also satisfy the inequality.

This illustration simplifies the concept of inequalities, showing both the boundary and all less-than values that meet the inequality's criteria.