Problem 105 - Introductory Algebra for College Students

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

Step 1: Analyze the algebraic expressions

First, observe the two expressions. The expression \(3+2x\) means 'Add 3 and two times some number x'. Whereas, in the expression \((3+2)x\), the parentheses indicate that 3 and 2 should be added together first, and then multiplied by x.

Step 2: Apply BODMAS rule on the second expression

In the expression \((3+2)x\), first add the numbers in the parentheses and then multiply it by x. So, it becomes \(5x\).

Step 3: Compare the two simplified expressions

On comparing \(3+2x\) and \(5x\), it's clear that these two are not the same.

Step 4: Validate the statement

The statement mentioned that the algebraic expressions \(3+2x\) and \((3+2)x\) do not mean the same thing. After simplifying and comparing the expressions, this statement has been validated as true.

Key Concepts

BODMAS rule

The BODMAS rule is a fundamental principle that helps us know the correct order to solve mathematical expressions. BODMAS stands for Brackets, Orders (like powers and square roots), Division and Multiplication, Addition and Subtraction.
It tells us what to do first when simplifying expressions:

• Start with any operations inside Brackets.
• Resolve Orders next (like exponents).
• Proceed with Division and Multiplication, from left to right. These operations are equal in priority.
• Finally, complete Addition and Subtraction from left to right.

The rule ensures consistency, whether you're working with numbers or algebraic expressions. Using BODMAS helps make calculations accurate and straightforward. Remember, brackets take the highest priority in expressions, ensuring the operations within them are completed first.

Parentheses in algebra

Parentheses play a crucial role in algebra as they indicate which part of the expression should be calculated first. They act as a signal, telling us to "do this part first!"
Using parentheses can change the meaning and the result of an expression significantly. For instance, compare these two expressions:

• Expression without parentheses: \(3 + 2x\)
• With parentheses: \((3 + 2)x\)

In \(3 + 2x\), the processes are: multiply 2 by \(x\) and add 3. In contrast, in \((3 + 2)x\), you first add 3 and 2, then multiply the result by \(x\).
This small change due to parentheses alters the whole expression's outcome. It helps in grouping terms you want to simplify first, or in showing that certain operations must be completed before others.
Pay attention to parentheses; they unlock strategies for more complex problem-solving by clearly defining the order of operations.

Simplifying expressions

Simplifying expressions is all about making them easier to work with by performing operations and combining like terms. This process doesn't change the value of the expression, just the way it looks.
To simplify, start by applying the BODMAS rule if necessary, especially if the expression has parentheses, exponents, or multiple operations. Simplification often involves:

• Combining like terms – These are terms that have the same variable and exponent. For example, \(2x\) and \(3x\) are like terms and can be added together to become \(5x\).
• Expanding expressions – This applies when using parentheses, such as \((3+2)x\), turning it into \(5x\).
• Canceling common factors in fractions or rational expressions when possible.

Simplifying can help in solving equations faster by reducing unnecessary steps. It's like tidying up; it might look simple but strategically organizes everything, paving the way for easier calculations and clearer understanding of the expression's behavior.