Problem 78 - Elementary and Intermediate Algebra

Question

Insert one of the symbols \(>,<,\) or \(=\) in the blank to make each statement true. $$ -0.666 \dots \quad \text{_____}\quad-0.6 $$

Step-by-Step Solution

Answer

\(-0.666\ldots < -0.6\)

Step 1: Understand the Values

First, observe the two numbers involved: - The first number is \(-0.666\ldots\),\ which is a repeating decimal equivalent to \(-\frac{2}{3}\).- The second number is \(-0.6\),\ which can also be written as \(-\frac{3}{5}\).

Step 2: Compare the Numbers

It's often easier to compare fractions when the denominators are the same. We need to check which is smaller or larger:- Convert \(-0.6\) to a fraction: \(-0.6 = -\frac{3}{5}\).- To compare \(-\frac{2}{3}\) and \(-\frac{3}{5}\), consider a common denominator: \(-\frac{2}{3} = -\frac{2 \times 5}{3 \times 5} = -\frac{10}{15}\) \(-\frac{3}{5} = -\frac{3 \times 3}{5 \times 3} = -\frac{9}{15}\).

Step 3: Analyze the Fractions

Let's analyze:- For the fractions \(-\frac{10}{15}\) (from \(-0.666\ldots\)) and \(-\frac{9}{15}\) (from \(-0.6\)), the larger magnitude exists in the numerically more negative number.- Since \(-\frac{10}{15}\) is more negative than \(-\frac{9}{15}\), we conclude \(-\frac{10}{15} < -\frac{9}{15}\).

Step 4: Write the Correct Symbol

Since \(-0.666\ldots\) translated to a more negative fraction than \(-0.6\), the correct inequality is:\(-0.666\ldots < -0.6\).

Key Concepts

Repeating Decimals

Repeating decimals are decimals that have one or more repeating numbers or sequences after the decimal point. For instance, in the problem we're looking at a decimal number like \(-0.666...\) where the '6' repeats infinitely. You might see repeating decimals written with a bar over the repeating digit(s), like this: \(-0.\overline{6}\). When dealing with repeating decimals, it can be very useful to convert them into fractions. This conversion often makes it easier to compare, calculate, and understand these numbers. In our example, \(-0.666...\) is equivalent to \(-\frac{2}{3}\). Once converted, you can handle these numbers more like ordinary fractions, which simplifies the process of comparing them to other numbers. Understanding the nature of repeating decimals helps clarify their values and provides deeper insights into their role in mathematics.

Inequalities

Inequalities are mathematical sentences expressing the relative size or order of two values. When you see symbols like \(>\), \(<\), or \(=\), these are used to compare numbers or expressions. In this particular exercise, we need to determine which of the two numbers \(-0.666...\) and \(-0.6\) is greater, and hence, which inequality symbol fits.To decide the correct inequality, we perform a comparison by converting both numbers to fractions (if needed) with common denominators. For this exercise:

• Convert \(-0.6\) to \(-\frac{3}{5}\).
• Express both fractions with a common denominator: \(-\frac{2}{3} = -\frac{10}{15}\) and \(-\frac{3}{5} = -\frac{9}{15}\).

By analyzing these, we determine that \(-\frac{10}{15}\) is less than \(-\frac{9}{15}\), leading us to the conclusion that \(-0.666... < -0.6\). This methodical approach ensures that the inequality reflects the correct relationship between the two numbers.

Equivalent Fractions

Fractions are equivalent if they express the same value or proportion, even though they might look different. When comparing fractions or repeating decimals, finding equivalent fractions often simplifies the process. For example, in converting the repeating decimal \(-0.666...\) to the fraction \(-\frac{2}{3}\), we create fractions that are easier to compare by using equivalent representations. For both \(-0.666...\) and \(-0.6\), converting them into fractions \(-\frac{2}{3}\) and \(-\frac{3}{5}\) respectively, and then converting to a common denominator helps highlight their size comparison:

• \(-\frac{2}{3} = -\frac{10}{15}\)
• \(-\frac{3}{5} = -\frac{9}{15}\)

Equivalent fractions allow us to perform direct comparisons, much like comparing whole numbers. Understanding how to convert and equate fractions provides a foundation for mastering number comparisons, ensuring clarity and precision in mathematical reasoning.