Problem 75 - Beginning and Intermediate Algebra

Question

Determine whether each statement is true or false. \(-5<-2\)

Step-by-Step Solution

Answer

The statement \(-5 < -2\) is true.

Step 1: Understand the inequality

The inequality given is \(-5 < -2\). This needs to be understood in terms of the number line or the properties of negative numbers.

Step 2: Compare the numbers

Identifying the position of the numbers on the number line: \(-5\) is further left on the number line compared to \(-2\), which means \(-5\) is a smaller number than \(-2\).

Step 3: Inferred conclusion

Since \(-5\) is to the left of \(-2\) on the number line, it confirms that \(-5 < -2\) is true.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero and often used to represent a deficit or a loss.
When working with negative numbers, it's helpful to remember a few key points:

• Negative numbers decrease further from zero.
• They are always found to the left of zero on a number line.
• A more negative number is always smaller than a less negative number (e.g., \(-5\) is smaller than \(-2\)).

These points are crucial in understanding inequalities involving negative numbers. When comparing negative numbers, it’s important to see how they align on the number line to determine which is smaller.

Using the Number Line

A number line is a visual representation of numbers in order, both positive and negative. Here’s how a number line helps us understand inequalities:

• Zero is placed at the center.
• Positive numbers are to the right of zero.
• Negative numbers are to the left of zero.
• The further left a number is, the smaller it is considered to be.

Let's visualize comparing \(-5\) and \(-2\):

• \(-5\) is five units to the left of zero.
• \(-2\) is two units to the left of zero.
• Clearly, \(-5\) is further to the left than \(-2\).

This confirms that \(-5 < -2\) because on the number line, being further left means being smaller.

Comparisons of Inequalities

Comparing inequalities involves understanding which number is greater or smaller.
Here’s a step-by-step approach to comparing negative numbers:

• Identify the numbers you need to compare.
• Use a number line if it helps to visualize their positions.
• Remember, the number further left (more negative) is always smaller.

Applying this to our example \(-5 < -2\):

• \(-5\) is compared to \(-2\).
• On the number line, \(-5\) is to the left of \(-2\).
• Therefore, \(-5\) is smaller than \(-2\), making the inequality true.

Understanding these comparisons helps in solving a variety of inequality problems, especially involving negative numbers.