Problem 45 - Basic College Mathematics

Question

Use \(=\) or \(\neq\) for \(\square\) to write a true sentence. $$ \frac{5}{2} \square \frac{17}{7} $$

Step-by-Step Solution

Answer

\r\frac{5}{2} > \frac{17}{7}

Step 1: Convert Fractions to Decimals

Convert \(\frac{5}{2}\) and \(\frac{17}{7}\) into decimal form to make comparisons easier. \(\frac{5}{2} = 2.5\) and \(\frac{17}{7} \approx 2.4286\).

Step 2: Compare Decimal Values

Compare the decimal values obtained: 2.5 and 2.4286. Since 2.5 is greater than 2.4286, we can determine the correct relational operator.

Step 3: Write the Inequality

Replace \(\box\) with the correct inequality operator based on the comparison: \(\frac{5}{2} > \frac{17}{7}\).

Key Concepts

converting fractions to decimals

Converting fractions to decimals is a crucial skill in comparing fractions. This involves dividing the numerator (top number) by the denominator (bottom number). For example, to convert the fraction \(\frac{5}{2}\):

- Divide 5 by 2, which equals 2.5.
- Hence, \(\frac{5}{2} = 2.5\).

Repeating the process for \(\frac{17}{7}\):
- Divide 17 by 7, yielding approximately 2.4286.
- Hence, \(\frac{17}{7} \approx 2.4286\).

These decimal representations make it easier to compare the fractions accurately.

comparing decimal values

Once we have the decimal equivalents of the fractions, comparing them becomes straightforward. Let's look at our values:

• 2.5 from \(\frac{5}{2}\)
• 2.4286 from \(\frac{17}{7}\)

Comparing 2.5 and 2.4286 involves checking which number is larger or smaller. Since 2.5 is greater than 2.4286, we conclude that

\[2.5 > 2.4286\]
Therefore, \(\frac{5}{2}\) is greater than \(\frac{17}{7}\).

Always remember to align the decimal places vertically if you are doing this by hand, and compare digit by digit from left to right.

inequality operators

Inequality operators help express the relationship between two values. The common inequality operators are:

• \(>\) : greater than
• \( < \) : less than
• \(=\) : equal to
• \(eq\): not equal to

In our problem, we need to decide the appropriate operator to make a true statement between the fractions \(\frac{5}{2}\) and \(\frac{17}{7}\).

After converting to decimals, we see 2.5 is greater than 2.4286. Thus, the correct inequality operator here is:

\[ \frac{5}{2} > \frac{17}{7} \]
This shows that 2.5 is indeed greater than 2.4286, and so the correct statement is:

\(\frac{5}{2} > \frac{17}{7}\).

Understanding these operators is essential for solving inequalities in more complex scenarios!