Problem 27 - Algebra and Trigonometry

Question

\(27-28=\) Place the correct symbol \((<,>, \text { or }=)\) in the space. \(\begin{array}{lllll}{\text { (a) } 3} & {\frac{7}{2}} & {\text { (b) }-3} & {-\frac{7}{2}} & {\text { (c) } 3.5} & {\frac{7}{2}}\end{array}\)

Step-by-Step Solution

Answer

(a) <, (b) >, (c) =.

Step 1: Convert Fractions to Decimals

First, we need to convert the fraction \( \frac{7}{2} \) to a decimal. To do this, divide 7 by 2. \( 7 \div 2 = 3.5 \).

Step 2: Compare Converted Values

Now, compare the given values: \(3\), \(-3\), and \(3.5\) with \(3.5\), which is the decimal equivalent of the fraction \(\frac{7}{2}\).

Step 3: Determine Symbol for (a) 3 and 3.5

Since 3 is less than 3.5, the correct symbol for \((a)\) is \(<\).

Step 4: Determine Symbol for (b) -3 and -3.5

Since -3 is greater than -3.5, the correct symbol for \((b)\) is \(>\).

Step 5: Determine Symbol for (c) 3.5 and 3.5

Since 3.5 is equal to 3.5, the correct symbol for \((c)\) is \(=\).

Key Concepts

Conversion of Fractions to Decimals

The process of converting fractions to decimals is fundamental in comparing these two numerical representations. To convert a fraction to a decimal, you simply divide the numerator (the top number) by the denominator (the bottom number). For example, to convert the fraction \( \frac{7}{2} \), you divide 7 by 2. This results in 3.5.

Let's look at some key points to remember when converting fractions:

• Make sure to divide the top number by the bottom number directly.
• If the division doesn't result in a whole number, express it as a decimal.

Converting fractions allows us to easily compare them to other decimals. Understanding this process will give you flexibility in solving problems and ensuring accuracy.

Inequality Symbols

Inequality symbols are crucial in expressing the relationship between two numbers. The primary symbols include \( <, >, \) and \( = \). Each symbol has a specific meaning:

• \( < \): Less than
• \( > \): Greater than
• \( = \): Equal to

When applying these symbols, it's important to understand the order and value of the numbers you are comparing. For instance, in the example problem, we placed the symbol between two numbers to represent their comparative size.

Remember:

• If the number on the left is smaller, use \( < \).
• If it is larger, use \( > \).
• If they are the same, use \( = \).

Practice using these symbols with various problems to become more familiar with their application.

Decimal Comparison

Decimal comparison involves evaluating decimal numbers to determine which is larger, smaller, or if they are equal. When comparing decimals, ensure each number is expressed in decimal form. Then, break the comparison down step-by-step:

• Line up the decimals by their decimal points.
• Compare each digit starting from the leftmost digit.
• Determine which number is greater or if they are equal.

For instance, in our example between 3 and 3.5, 3 is less than 3.5 when placed in a comparison, making it smaller and hence, the symbol \( < \) is used. On comparing negatives like -3 and -3.5, -3 is greater as it’s less negative, so \( > \) is used.

Practicing these comparisons will develop a keen sense of numerical relationships and accuracy in mathematical problems.