Problem 62 - Introductory and Intermediate Algebra for College Students 4th

Question

Insert either \(<\) or \(>\) in the shaded area between each pair of numbers to make a true statement. $$-\frac{\pi}{2} \quad\square-2.3$$

Step-by-Step Solution

Answer

-π/2 is greater than -2.3, so the correct symbol to insert is \'\>\'.

Step 1 - Convert π to a decimal

π is approximately equal to 3.14. Therefore, -π/2 is approximately equal to -3.14/2.

Step 2 - Simplify decimal

Divide -3.14 by 2 to get about -1.57.

Step 3 - Compare the two numbers

Compare -1.57 and -2.3. A number is greater if it is closer to zero on the number line. Therefore, -1.57 is greater than -2.3.

Step 4 - Insert symbol

Consequently, the correct symbol to insert is \'\>\'.

Key Concepts

Number Line

A number line is a simple visual tool that helps us understand the position of numbers. It is a straight line where every point represents a real number.
To the right of zero, numbers are positive and grow larger as you move further right. To the left, numbers are negative and decrease.

• Numbers closer to zero are always greater than those further away, regardless of whether they are positive or negative.
• This means that on the number line, \(-1.57\) is closer to zero than \(-2.3\), making \(-1.57\) greater.

Using this concept helps us visually compare and better understand the relationship between numbers.

Decimal Approximation

Decimal approximation involves converting numbers, like fractions or irrational numbers, into decimal form to make them easier to compare. For \(-\frac{\pi}{2}\), we approximate \(\pi\) as \(3.14\).

• Dividing \(-3.14\) by \(2\) yields approximately \(-1.57\), which is easier to work with than its original fraction form.
• This technique allows us to more easily compare it to other decimals, like \(-2.3\).

Approximating helps simplify calculations and understand the relative magnitude of different numbers.

Comparison of Real Numbers

Comparing real numbers can be a bit tricky, but with practice, it becomes straightforward. We look at their position relative to zero and each other.

• Smaller negative numbers (closer to zero) are actually larger. That's because they represent less negative value.
• In our example, \(-1.57\) is a smaller negative number compared to \(-2.3\), meaning \(-1.57\) is greater.

Understanding these principles allows for accurate comparison and helps avoid the common pitfalls of comparing negative values.