Problem 62 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

\(-\frac{\pi}{2} > -2.3\)

Step 1: Convert fraction to decimal

The first thing to do is to convert the fractional representation, \(-\frac{\pi}{2}\), to a decimal. Since the value of \(\pi\) is approximately 3.1416, \(-\frac{\pi}{2}\) is approximately -1.5708.

Step 2: Compare the numbers

Now it's time to compare the two numbers to see which is greater. When we compare -1.5708 (which is \(-\frac{\pi}{2}\)) and -2.3, we can see that -1.5708 is greater than -2.3. This is because -1.5708 is to the right of -2.3 on the number line. Remember, a number is 'greater' if it is to the right of another number on the number line.

Step 3: Insert the correct symbol

So, since \(-\frac{\pi}{2}\) is greater than -2.3, the symbol to put in the box is \(>\). This makes the inequality \(-\frac{\pi}{2} > -2.3\) true.

Key Concepts

Comparing Numbers

Understanding the concept of comparing numbers is fundamental in algebra, especially when dealing with inequalities. An inequality indicates that two values are not equal and that one is greater or smaller than the other. We often demonstrate this using symbols like '<' for less than and '>' for greater than.

For example, if we have two numbers, A and B, and if A appears to the left of B on a number line, it means A is less than B (A < B); conversely, if A is to the right of B, A is greater than B (A > B). In our exercise, we're asked to compare \( -\frac{\pi}{2} \) and -2.3—the position of these numbers on the number line guides us. Negative numbers can be tricky, but remember, the more negative a number, the less its value. In essence, -1 is greater than -2 because it is closer to zero on the number line.

Fraction to Decimal Conversion

Converting fractions to decimals is a vital skill in mathematics as it allows for easier comparison and computation. To convert a fraction to a decimal, simply divide the numerator by the denominator. In our specific scenario, we have the fraction \( -\frac{\pi}{2} \).

We start by understanding that \( \pi \) is an irrational number approximately equal to 3.1416. Division by 2 gives us 1.5708, and because our fraction is negative, we get approximately -1.5708. Converting fractions to decimals is especially useful when the fractions involve constants like \( \pi \) because it simplifies the number to a form that can be compared to others on a number line or in equations. Always round to a reasonable number of decimal places for practical comparison purposes.

Inequalities with Pi

Dealing with inequalities involving \( \pi \) adds a layer of complexity due to the transcendental nature of \( \pi \)—it's a non-repeating, non-terminating decimal. Yet, in many contexts, \( \pi \) can be approximated to 3.1416 (or even simply 3.14 for basic calculations) which can then be used in operations such as division when \( \pi \) is part of a fraction.

For instance, when comparing \( -\frac{\pi}{2} \) with another decimal, the conversion to decimal form is pivotal. In this case, once \( -\frac{\pi}{2} \) is converted to -1.5708, the sign of the inequality can be determined by analyzing which number is greater or lesser, factoring in that \( \pi \) is essentially a constant value. This approach makes it intuitive to solve inequalities that feature \( \pi \) or any other irrational numbers.