Problem 16 - Elementary Algebra

Question

For the following problems, write the appropriate relation symbol \((=,<,>)\) in place of the \(*\). $$ 3(1.06+2.11) * 4(11.01-9.06) $$

Step-by-Step Solution

Answer

Answer: The correct relation symbol is ">". The complete inequality is: 3(1.06 + 2.11) > 4(11.01 - 9.06).

Step 1: Simplify the left side of the inequality

First, we need to simplify the left side of the inequality. We will do this by performing the operations inside the parentheses and then multiplying by the number outside the parentheses: $$ 3(1.06 + 2.11) = 3(3.17) = 9.51 $$

Step 2: Simplify the right side of the inequality

Next, we will simplify the right side of the inequality in the same manner: $$ 4(11.01 - 9.06) = 4(1.95) = 7.80 $$

Step 3: Compare the simplified expressions

Now that we have the simplified values for both sides of the inequality, we can compare them to determine the correct relation symbol: $$ 9.51 * 7.80 $$ Since \(9.51 > 7.80\), the correct relation symbol is \(>\):

Step 4: Final Answer

Therefore, the correct inequality is: $$ 3(1.06 + 2.11) > 4(11.01 - 9.06) $$

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental part of working with inequalities and equations. This process involves reducing expressions to a simpler form in order to make calculations easier and more comprehensible. In the given exercise, we encountered expressions such as \(3(1.06 + 2.11)\) and \(4(11.01 - 9.06)\). To simplify them, we need to perform the arithmetic operations inside the parentheses first. This step is essential as it helps in breaking down complex expressions into manageable calculations.

Here's how we do it:

• First, resolve the operations inside the parentheses: \(1.06 + 2.11 = 3.17\) and \(11.01 - 9.06 = 1.95\).
• Next, multiply the results by the numbers outside the parentheses: \(3 \times 3.17 = 9.51\) and \(4 \times 1.95 = 7.80\).

These steps result in simplified forms. Through simplification, we can easily compare two expressions and determine their relationship.

Order of Operations

Understanding the order of operations is crucial when simplifying expressions, especially in inequalities. The order of operations is a set of rules that determines the correct sequence in which to solve parts of a mathematical expression. Often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), this rule is pivotal for ensuring accurate calculations.

In our exercise, the order of operations dictated that we first handle the calculations inside the parentheses before proceeding to multiplication.

• Start with Parentheses: Calculate \(1.06 + 2.11\) and \(11.01 - 9.06\).
• Proceed to Multiplication: Multiply the results by the respective numbers outside the parentheses: \(3\) and \(4\).

By following these steps, we preserve mathematical accuracy and arrive at the correct relationship between simplified expressions in the inequality.

Mathematical Symbols

Mathematical symbols play a key role in expressing relationships between numbers and operations. In inequalities specifically, symbols like \(=, <,\) and \(>\) are used to show how one expression compares to another. These symbols provide a concise and universal way to communicate mathematical relationships.

Let's look at how these symbols are used:

• The equals symbol \(=\) indicates that two quantities are identical.
• The less than symbol \(<\) shows that the quantity on the left is smaller than the one on the right.
• The greater than symbol \(>\) is used when the quantity on the left is larger than the one on the right.

In the example provided, we determined that \(9.51\) is greater than \(7.80\). Therefore, the correct inequality symbol to use between the two simplified expressions is \(>\), illustrating their relationship.