Question

Mark the answers true or false as follows: A. True B. False \((1+x-1)\) is always equal to \(x\).

Short answer

True.

Step-by-step solution

Simplify the Expression

First, let's simplify the expression \((1 + x - 1)\). To simplify, combine like terms.\(1\) and \(-1\) are like terms, and their sum is \(0\). Thus, the expression simplifies to \(x\).

Analyze the Expression

We need to check if \((1 + x - 1)\) is equal to \(x\) for all values of \(x\). Since we simplified the expression to \(x\), we can see that the original expression is indeed equal to \(x\) for any value of \(x\).

Determine the Truth Value

Since the simplified expression \((1 + x - 1)\) equals \(x\) for any value of \(x\), the statement is true. Therefore, we mark the answer as A. True.

Key concepts

Simplification

Simplification in algebra is the process of reducing an algebraic expression to its most basic form. This involves combining like terms, removing unnecessary components, and making the expression easier to work with.

• Combining Like Terms: When simplifying expressions, look for terms that have the same variable to the same degree. For example, in the expression \(1+x-1\), the numbers \(1\) and \(-1\) are like terms.
• Simplification Steps: Identify these terms and combine them to change the expression \(1+x-1\) to \(x\). The numbers \(1\) and \(-1\) add up to zero, leaving us with \(x\).

Simplifying makes it easier to analyze and work with algebraic equations because it allows you to see the core variable expressions without clutter. It is a foundational skill necessary for solving more complex algebraic problems.

Expression Analysis

Expression analysis involves examining an algebraic expression to understand its structure and behaviour without performing calculations at every step.

• Understanding Expressions: When analyzing the expression \(1 + x - 1\), we want to determine if, after simplification, it behaves consistently in various conditions.
• Consistent Result: After simplifying to \(x\), we need to ask whether this result holds true for all values of \(x\). In this case, it does, since whatever value \(x\) takes, the simplified expression remains the same.

Through expression analysis, we ensure that regardless of the specific numbers involved, the general form and properties of the expression do not change. This is crucial for proving that algebraic operations performed on expressions are correct.

Truth Values in Algebra

Truth values in algebra tell us whether statements or equations hold true under specified conditions.

• Examining Statements: After simplifying and analyzing the expression to see if \(1 + x - 1 = x\), we look at whether this equation always holds true.
• Identifying True Statements: The simplified and analyzed expression shows that the statement is true uniformly across all cases. This means that \(1 + x - 1\) is always \(x\), making the claim valid.

Assigning truth values helps in understanding whether our simplifications and analyses lead to universally applicable results. This ensures that we can rely on them in various mathematical contexts without needing further testing. Marking the answer as true in this case is a verification of the consistent nature of the equation across all possible instances.