Problem 73 - Elementary and Intermediate Algebra

Question

Insert one of the symbols \(>,<,\) or \(=\) in the blank to make each statement true. \(|-\frac{15}{2}| {}\quad 7.5\)

Step-by-Step Solution

Answer

The correct symbol is \(=\).

Step 1: Understand Absolute Value

The absolute value of any number is the non-negative value of that number without regard to its sign. For example, the absolute value of a negative number is a positive number. Here, we need to find the absolute value of \(-\frac{15}{2}\).

Step 2: Calculate the Absolute Value

To calculate the absolute value of \(-\frac{15}{2}\), remove the negative sign. Therefore, the absolute value is \(\frac{15}{2}\).

Step 3: Convert Fraction to Decimal

Convert \(\frac{15}{2}\) to a decimal by dividing 15 by 2, which equals 7.5.

Step 4: Insert the Correct Symbol

Now we have \(7.5\) on both sides: \(|-\frac{15}{2}| = 7.5\). Since both values are equal, we insert the symbol \(=\) to make the statement true.

Key Concepts

Fraction to Decimal Conversion

Converting a fraction to a decimal is an important skill in mathematics. It involves dividing the numerator (top number) by the denominator (bottom number). This process is especially useful for making comparisons, as working with decimals can be simpler than fractions.

For instance, in the exercise provided, we converted the fraction \(\frac{15}{2}\) to a decimal. This was done by dividing 15 by 2, which equals 7.5.

Here are some key steps to follow when converting fractions to decimals:

• Identify the fraction you need to convert.
• Divide the numerator by the denominator using long division if needed.
• Place the result in decimal form for easier computation and comparison.

This method allows us to quickly and accurately change a fraction into a more easily understandable decimal form.

Inequality Symbols

Inequality symbols help express the relative size or relationship between two numbers or expressions. There are several symbols that you should be familiar with:

• \(>\): means "greater than"
• \(<\): means "less than"
• \(=\): means "equal to"

These symbols are crucial for forming correct mathematical statements and arguments. In the provided exercise, understanding these symbols allowed us to determine the relationship between \(|-\frac{15}{2}|\) and 7.5. Through computation, once both values were found to be 7.5, the correct symbol was \(=\), showing equality.

When using these symbols:

• Compare the two expressions or numbers directly.
• Consider each side of the equation independently, especially if further calculations are needed.
• Select the correct symbol based on the relative size of the numbers.

Understanding how to use inequality symbols correctly allows for proper mathematical reasoning and logic.

Solving Algebra Problems

Solving algebra problems often involves combining various mathematical techniques and knowledge areas. These range from arithmetic operations to understanding and applying mathematical symbols and terms correctly.

An algebra problem typically requires you to:

• Understand the problem statement and what is being asked.
• Perform any arithmetic operations required, such as addition, subtraction, multiplication, division, or finding the absolute value.
• Use mathematical logical reasoning to decide on the appropriate actions and symbols to ensure the statement or equation holds true.

In our specific exercise, correctly solving the problem required:

• Finding the absolute value, \(|-\frac{15}{2}|\).
• Converting the resulting fraction to a decimal, equating its value to the reference number.
• Determining and applying the right inequality or equality symbol to complete the statement.

This systematic approach ensures solutions are both accurate and comprehensible, simplifying the problem-solving process and minimizing errors.