Problem 105 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The algebraic expressions \(3+2 x\) and \((3+2) x\) do not mean the same thing.

Step-by-Step Solution

Answer

The statement 'The algebraic expressions \(3+2 x\) and \((3+2) x\) do not mean the same thing.' is true.

Step 1: Simplify the first expression

The first expression is \(3+2 x\). According to the order of operations (BODMAS/PEMDAS), there are no brackets or orders. So, the expression remains the same, i.e., \(3+2x\).

Step 2: Simplify the second expression

The second expression is \((3+2) x\). According to the order of operations (BODMAS/PEMDAS), calculations inside brackets should be carried out first. So, \(3+2\) inside the brackets equals to 5. Therefore, the simplified form of this expression is \(5x\).

Step 3: Compare both expressions

Now, comparing the simplified forms of both expressions, \(3+2x\) is not equal to \(5x\), which signifies that the two expressions do not, in fact, mean the same thing.

Key Concepts

Order of Operations

When working with algebraic expressions, understanding the order of operations is crucial. This sequence dictates the correct order in which to perform calculations to obtain the correct result. In mathematics, this is most often referred to as BODMAS or PEMDAS, acronyms that remind us of the operational hierarchy:

• Brackets/Parentheses
• Orders/Exponents (such as squares and cubes)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

In the exercise example, the expression (3+2)x illustrates the use of brackets which need to be simplified first. This is because brackets have the highest priority in the order. Therefore, we calculate 3+2 to get 5, before proceeding with multiplication by x. By following these rules, you can solve expressions accurately, avoiding common errors and confusion.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form. This often involves combining like terms and executing the operations as laid out by the order of operations.
In the given exercise, the first expression is \(3 + 2x\). There are no parenthetical groupings or exponents present, so it is already in its simplest form.
The second expression, \((3+2)x\), requires simplification within the brackets (\(3+2 = 5\)). After simplifying inside the brackets, we multiply out, yielding the final expression \(5x\). Simplifying ensures expressions are presented in the most straightforward way, reducing errors and aiding clarity.

Mathematical Comparison

Mathematical comparison involves examining two or more expressions to determine their equivalency or difference. It's an essential skill in algebra to understand whether two expressions are equal or if one is greater or less than the other.
In the exercise task, you compare \(3+2x\) and \(5x\). Once simplified, it's clear that these expressions are not equal.

• The first expression, \(3+2x\), represents a sum of \(3\) and twice the value of \(x\).
• The second expression, \(5x\), represents five times the value of \(x\).

This comparison highlights a crucial aspect of algebra: structure matters. Even slight differences in an expression's structure or order can lead to different numeric values.Understanding these types of comparisons aids in ensuring mathematical accuracy and solving algebraic problems with confidence.