Problem 34 - Prealgebra

Question

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

Step-by-Step Solution

Answer

Shade from -5 to the left on the number line with a closed circle at -5.

Step 1: Identify the Inequality Type

The inequality is \( x \leq -5 \), which involves a less than or equal to comparison.

Step 2: Determine the Graphical Representation of the Inequality

Since \( x \leq -5 \) includes \(-5\), we will use a closed (filled-in) circle at \(-5\) on the number line.

Step 3: Shade the Appropriate Region

Shade all values to the left of \(-5\) on the number line, indicating that \( x \) can be any value less than or equal to \(-5\).

Step 4: Draw the Number Line

Draw a horizontal line with a point at \(-5\) filled in (indicating inclusion). Then, draw an arrow extending from the point \(-5\) to the left, representing all values less than \(-5\). This completes the graph of the inequality.

Key Concepts

Number Line

A number line is a simple visual tool to help understand the placement and relationship of numbers. Imagine a horizontal line marked with numbers in a sequence, typically at equal intervals.
Using a number line, you can easily see where a number appears in relation to other numbers.

• Positive numbers are usually placed to the right of zero.
• Negative numbers are shown to the left of zero.

The number line serves as a foundational tool in mathematics for comparing numbers, understanding absolute values, and solving inequalities.
It's a visual method that allows for quick comprehension of relationships between values, which is essential in graphing inequalities.

Representation of Inequalities

Representing inequalities on a number line involves understanding inequality symbols and their implications. The basic inequality symbols are:

• \( < \) means "less than"
• \( \leq \) means "less than or equal to"
• \( > \) means "greater than"
• \( \geq \) means "greater than or equal to"

When graphing inequalities, these symbols determine how we show the points on a number line.
For instance, a closed circle is used to represent "less than or equal to" or "greater than or equal to" because these include the specific number itself.
Open circles, on the other hand, indicate strictly "less than" or "greater than" without including the number itself.
After deciding on the circle type, the next step is to shade the number line in the direction of the inequality, showing the range of possible solutions.

Inequalities in Prealgebra

Prealgebra introduce students to inequalities, providing a stepping stone to understanding algebraic concepts. At this stage, students learn:

• How to solve simple inequalities
• Ways to graph them on a number line
• Understanding the meaning behind the inequality symbols

Inequalities challenge students to think beyond exact numbers and consider ranges of solutions.
This broadens their comprehension of problem-solving in mathematics compared to strict equality.
In prealgebra, focusing on visual methods like a number line makes it easier for students to grasp abstract concepts, reinforcing their confidence in mathematics as they continue their education.