Problem 61 - Introductory Algebra for College Students

Question

Insert either \(<\) or \(>\) in the shaded area between each pair of numbers to make a true statement. $$-\pi \square-3.5$$

Step-by-Step Solution

Answer

-\(\pi\) > -3.5

Step 1: Analyze the given numbers

The given numbers are \(-\pi\) and -3.5. \(-\pi\) is approximately equal to -3.14159.

Step 2: Insert the correct inequality sign

Since -3.14159 is greater than -3.5 on a number scale, the correct inequality sign to insert between \(-\pi\) and -3.5 is \(>\). So, -\(\pi\) > -3.5 is the correct inequality.

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero. They are often used to represent values below a certain reference point, like temperatures below freezing, or depths below sea level. In mathematics, they are shown with a minus sign (-). For example, -3 is three units below zero.

When comparing negative numbers on a number line, keep in mind:

• A number is more negative if it is further left on the line.
• If two negative numbers are compared, the one closer to zero is larger. For instance, -1 is greater than -3 because it is closer to zero.

Understanding negative numbers helps work with inequalities involving negative values. In the exercise, \(-\pi\) is compared to -3.5, and because -3.14159 is closer to zero than -3.5, \(-\pi\) is indeed greater.

Number Line

A number line is a visual representation essential for understanding the position and value of numbers, both positive and negative. It is a straight line with numbers at intervals, and zero as the reference point usually in the center.

When using a number line:

• Moving right represents increasing values, while moving left indicates decreasing values.
• Negative numbers are placed on the left side of zero.
• Comparing values is easier with a number line view because their positions tell us which is larger or smaller.

In our original step-by-step solution, utilizing the number line helps in comparing \(-\pi\) with -3.5, allowing us to see that \(-\pi\)'s approximate value (-3.14159) is closer to zero and thus greater than -3.5.

Pi Approximation

Pi (\(\pi\)) is a special mathematical constant representing the ratio of a circle's circumference to its diameter. It is an irrational number, meaning it cannot be precisely written as a simple fraction, and it has a non-repeating decimal that goes on forever.

Pi is approximately 3.14159, though for most calculations, 3.14 is frequently used for simplicity.
When working with negative values, \-\(\pi\)\ becomes approximately -3.14159. Understanding this approximation is crucial in exercises like the one provided. Knowing that \(-\pi\) is about -3.14159, one can determine its position relative to other negative numbers like -3.5. Hence, comparing these gives clarity to whether \(-\pi\) is greater or lesser than the number in question.