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Chapter 1: Problem 40

Simplify the given algebraic expressions. $$-\\{-[-(x-2 a)-b]-(a-x)\\}$$

Short Answer

Expert verified
The simplified expression is \(-a + b\).

Step by step solution

01

Remove Inner Parentheses

Start by looking at the expression \(-\{-[-(x-2a)-b]-(a-x)\}\). Identify the innermost parentheses first which is \(-(x-2a)\). Removing the negative sign, this part becomes \(-x + 2a\). Now the expression inside the first square bracket becomes \(-x + 2a - b\).
02

Simplify the Intermediate Expression

Substitute the simplified expression \(-x + 2a - b\) back into the original expression: \(-\{-[-x + 2a - b]-(a-x)\}\). Next, remove more parentheses by simplifying \(-(a-x)\), which becomes \(-a + x\). Now the expression in the square brackets is \(-x + 2a - b - a + x\).
03

Combine Like Terms

Inside the square brackets, combine like terms: \(-x + x + 2a - a - b\). This simplifies to \(a - b\). The expression now updates to \(-\{a - b\}\).
04

Final Simplification

To finish simplifying, remove the last set of braces by applying the negative sign, which yields \(-a + b\). This is the fully simplified expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parentheses Removal
Removing parentheses is an initial step in simplifying algebraic expressions. Parentheses, brackets, or braces show which operations to perform first in an expression. Think of them as containers that group terms together. When simplifying, start by removing the innermost parentheses and work outward.
  • Begin with the expression \(-\{-[-(x-2a)-b]-(a-x)\}\), and identify the inner parentheses: \(-(x-2a)\).
  • To remove the negative sign, distribute it across the terms within the parentheses, turning \(x-2a\) into \(-x+2a\).
  • Replace the simplified expression back, leading to: \(-x + 2a - b\).
After these changes, the expression within the brackets becomes more manageable, allowing you to proceed to the next steps.
Combining Like Terms
Combining like terms is critical for simplifying expressions effectively. Like terms are those that have the same variable raised to the same power. You can add or subtract their coefficients. In the expression simplification, look for similarities within the terms.
  • After removing more parentheses, the expression \(-x + 2a - b - a + x\) appears in the brackets.
  • Identify and combine the compatible terms: \(-x\) and \(+x\) cancel each other out, simplifying to \(2a-a\), and leaving \(-b\).
  • The resulting expression becomes simply \(a - b\).
By systematically combining like terms, the expression rapidly shortens, making it easier to handle in subsequent steps.
Simplification Steps
Simplification is the process of making an expression easier to understand or work with. It involves performing operations like combining like terms and removing unnecessary parts. Once all parentheses are dealt with, finalize the simplification.
  • The brackets and operations within them are already simplified to \(a - b\).
  • Apply a negative sign from the remaining braces, changing \(a - b\) to \(-a + b\).
This is the final simplified form of the given expression. Simplification reduces complexity and reveals core relationships between terms, which is valuable in solving equations or understanding algebraic connections.

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