Problem 64 - Intermediate Algebra

Question

Insert either \(a<\) or \(a>\) symbol to make a true statement. $$ -6.07 \quad-\frac{17}{6} $$

Step-by-Step Solution

Answer

The correct symbol is <, so \\(-6.07 < -\frac{17}{6}\\).

Step 1: Convert fractions to decimals

To compare the two numbers given, it's helpful to convert the fraction to a decimal. The fraction \(-\frac{17}{6}\) can be converted by dividing -17 by 6. \(-\frac{17}{6} = -2.8333...\) (repeating).

Step 2: Compare the decimal numbers

Now compare \(-6.07\) with \(-2.8333...\). A number is smaller if it is more negative. Therefore, \(-6.07\) is less than \(-2.8333...\), since \(-6.07\) is further left on the number line.

Step 3: Insert the correct symbol

Since \(-6.07\) is less than \(-2.8333...\), we use the symbol \(<\) which stands for 'less than'. Therefore, the correct statement is: \(-6.07 < -\frac{17}{6}\).

Key Concepts

Comparing Decimals

When comparing decimals, the goal is to determine which of the two numbers is larger or smaller. This doesn't just apply to positive numbers; understanding negative decimals is equally important.
Steps for Comparing Decimals:

• Align the numbers by their decimal points to ensure an accurate comparison.
• Start comparing from the leftmost digit. For negative numbers, remember that a number is considered smaller the more negative it is.
• If the integer part (whole number part) is the same, compare the decimal parts digit by digit.

Consider the numbers \( -6.07 \) and \( -2.8333.... \). With both numbers being negative, \( -6.07 \) is more negative than \( -2.8333... \). Therefore, it is smaller.

Converting Fractions to Decimals

Fractions can often be perplexing, but converting them to decimals can make comparison and other operations more straightforward. The conversion involves division, which turns a fraction into a decimal format.
Steps to Convert Fractions to Decimals:

• Take the numerator (the top number) and divide it by the denominator (the bottom number).
• Perform the division to as many decimal places as needed, especially if the decimal is repeating or terminating.
• If the decimal repeats, note it with a line or ellipsis after the last repeating digit.

For instance, converting the fraction \( -\frac{17}{6} \) is done by dividing -17 by 6. The result \( -2.8333... \) is a repeating decimal which makes it easy to compare with other decimals.

Number Line

A number line is a visual aid that can help effectively compare numbers, even with decimals or fractions. It horizontally represents numbers, allowing you to see which numbers are smaller or larger based on their position from left to right.
Using Number Lines:

• The left side of the number line indicates smaller numbers or more negative numbers.
• The right side represents larger numbers or more positive numbers.
• When positioning decimals, it helps to think about their value in comparison to familiar reference points like 0, -1, 1, etc.

In our example, \( -6.07 \) is placed further left on the number line compared to \( -2.8333... \), visually confirming that it is indeed the smaller number. Using a number line can simplify concepts that might seem abstract at first.