Problem 47 - Prealgebra

Question

Replace each \(\circ\) with \(<,>,\) or \(=\) to make a true statement. $$-5 . \overline{3} \circ 5.333 \ldots$$

Step-by-Step Solution

Answer

-5.3333... < 5.3333...

Step 1: Understand the Problem

We need to compare two numbers: \(-5.\overline{3}\) and 5.333... by replacing the \(\circ\) symbol with either \(<\), \(>\), or \(=\) to form a true inequality or equation.

Step 2: Analyze the Numbers

The first number given is \(-5.\overline{3}\), which is a repeating decimal. It represents \(-5.3333\ldots\). The second number is \(5.333\ldots\), which is positive and equals \(5.3\) repeating.

Step 3: Compare Negatives and Positives

Since \(-5.3333\ldots\) is a negative number and \(5.333\ldots\) is a positive number, any negative number is always less than any positive number.

Step 4: Write the Inequality

Based on the comparison in Step 3, we conclude that \(-5.3333\ldots < 5.3333\ldots\). So, we replace \(\circ\) with \(<\).

Key Concepts

Repeating Decimals

Repeating decimals can be a bit confusing at first, but they become much clearer once we understand what they represent. When you see a number like \(-5.\overline{3}\), the line above the 3 indicates that this digit repeats indefinitely. This means the decimal is actually \(-5.33333\ldots\). To grasp repeating decimals:

• Look for the repeating digit, marked with a bar or dots.
• Understand that this digit continues infinitely.

Another example is the number 5.333..., where the 3 is repeating, marked by the ellipses (...). This can also be written as \(5.\overline{3}\). Understanding these repeating patterns helps us represent and compare these decimals accurately.

Negative Numbers

Negative numbers are numbers that are less than zero. They provide us with a way to express values below a point of reference, like sea level. In the realm of mathematics, dealing with negatives is crucial:

• They allow representation of debts or temperatures below zero.
• They follow specific rules in operations, such as when multiplying, two negatives yield a positive.

When comparing numbers, any negative number is always less than any positive number. In this context, the number \(-5.3333\ldots\) is negative, meaning it inherently possesses a lower value compared to any positive figure, including 5.333..., enabling us to make quick comparisons.

Positive Numbers

Positive numbers represent quantities more than zero. They are straightforward and can be found on the right side of the number line. Positive numbers are crucial in daily life as they:

• Indicate gains, like profits or numbers of items.
• Stand as metrics for distances or heights above sea level.

When compared to negative numbers, positive numbers are always greater. In this exercise, we have the positive number 5.333... that stands above the negative number \(-5.3333\ldots\) in terms of value. Recognizing the distinction between negative and positive numbers simplifies inequality evaluation.