Problem 15 - Elementary Algebra

Question

For the following problems, write the appropriate relation symbol \((=,<,>)\) in place of the \(*\). $$ 9[4+3(8)] * 6[1+8(5)] $$

Step-by-Step Solution

Answer

Question: Determine the correct relation symbol for the following expressions: \(9[4 + 3(8)] * 6[1 + 8(5)]\). Answer: The correct relation symbol between the two expressions is \(>\) (greater than), making the final expression \(9[4 + 3(8)] > 6[1 + 8(5)]\).

Step 1: Simplify the first expression

Start by simplifying the expression inside the brackets and then apply the correct order of operations. $$ 9[4+3(8)] = 9[4 + 24] = 9[28] $$ Now, multiply the result inside the brackets by 9: $$ 9[28] = 252 $$ The first expression simplifies to \(252\).

Step 2: Simplify the second expression

Again, start by simplifying the expression inside the brackets and then apply the correct order of operations. $$ 6[1 + 8(5)] = 6[1 + 40] = 6[41] $$ Now, multiply the result inside the brackets by 6: $$ 6[41] = 246 $$ The second expression simplifies to \(246\).

Step 3: Find the appropriate relation symbol

Now, we will compare the results of both expressions to determine which relation symbol to use. Since \(252 > 246\), the correct relation symbol is \(>\) (greater than).

Step 4: Write the final answer

Replace the \(*\) with the correct relation symbol (\(>\)) between the two expressions: $$ 9[4 + 3(8)] > 6[1 + 8(5)] $$

Key Concepts

Understanding Inequalities

Inequalities are mathematical statements that compare two values or expressions. They help us understand how one number relates to another. In this exercise, we were asked to replace the asterisk (\(*\)) with a relational symbol like\(=\), \(<\), or \(>\). These symbols indicate whether the expressions are equal, less than, or greater than one another, respectively.
Understanding inequalities is crucial since they show relational differences rather than equalities. For example:

• \(a = b\) means that \(a\) is equal to \(b\).
• \(a < b\) signifies that \(a\) is less than \(b\).
• \(a > b\) indicates that \(a\) is greater than \(b\).

In the context of this problem, the two expressions were compared and found to satisfy the inequality \(252 > 246\). So, the relational symbol \(>\) was used.

Expression Simplification

Simplifying mathematical expressions is a fundamental skill that involves reducing expressions to their simplest form. This makes them easier to understand and compare.
The process typically starts by simplifying inside any parentheses or brackets. This is followed by applying the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
In our exercise, we simplified \(9[4+3(8)]\) and \(6[1+8(5)]\) in a step-by-step manner:

• First, evaluate the bracketed expressions: \(4 + 3(8)\) becomes \(4 + 24\), and \(1 + 8(5)\) becomes \(1 + 40\).
• Next, add within the brackets: \(9[28]\) and \(6[41]\).
• Finally, perform the multiplication: \(9 imes 28 = 252\) and \(6 imes 41 = 246\).

Once simplified, it becomes straightforward to compare the final values and determine the relationship between them.

The Role of Mathematical Symbols

Mathematical symbols play an essential role in conveying complex ideas succinctly. They allow mathematicians and students alike to express concepts without lengthy explanations.
In this exercise, three key symbols were involved: relational symbols \(=\), \(<\), and \(>\). These symbols communicate the results of the expressions in a concise manner.
Understanding these symbols is crucial to grasping mathematical concepts more broadly:

• \(=\) is the equality symbol, indicating that the two sides of the equation are the same.
• \(<\) is the less-than symbol, showing that the number on the left is smaller than the number on the right.
• \(>\) is the greater-than symbol, signifying that the number on the left is larger than the number on the right.

Correctly identifying these symbols helps us understand the relationship between values and ensures that we interpret mathematical expressions accurately.