Problem 76 - Algebra for College Students

Question

Explain how to simplify the expression $$ 7 x-[3 x-(2 x-4)+2]-x $$

Step-by-Step Solution

Answer

The simplified expression is \(5x - 6\).

Step 1: Identify and Organize

First, clearly write down and organize the expression \[7x - [3x - (2x - 4) + 2] - x\]Don't remove any parentheses yet but note where the grouping symbols are.

Step 2: Simplify the Innermost Parentheses

Within the bracket, identify and simplify the innermost parentheses:\[2x - 4\]This can't be simplified further, so leave it as is while removing the parentheses in the next steps.

Step 3: Expand the Brackets

Focus on expanding the brackets step by step. Begin with the innermost expression:\[ 3x - (2x - 4) + 2 \]Distribute the negative sign through the parentheses:\[3x - 2x + 4 + 2 \]Combine like terms:\[x + 6\]

Step 4: Substitute and Simplify the Expression

Replace the simplified expression back into the original structure:\[7x - [x + 6] - x\]Distribute the negative sign through the bracket:\[7x - x - 6 - x \]Combine all terms together to simplify:\[ 7x - x - x - 6 = 5x - 6\]

Step 5: Double-Check Your Work

Review the simplified expression:\[5x - 6\]Ensure all steps are properly evaluated and terms are combined correctly. Confirm that there are no more like terms to combine.

Key Concepts

Parentheses Removal

When working with algebraic expressions, removing parentheses is a crucial step in simplification. Parentheses are used to indicate which operations should be performed first. Removing them correctly ensures the expression maintains its intended meaning.

To effectively remove parentheses:

• Begin with the innermost parentheses first, as operations inside should be prioritized.
• Be cautious of any negative signs or coefficients before the parentheses, as they affect all terms inside upon removal.
• Simplify the expression inside the parentheses as much as possible beforehand, reducing the steps required later.

In our example, removing parentheses involved acknowledging the negative sign before the innermost group:\[3x - (2x - 4)\]Distributing this negative sign transforms the expression, indicating its importance right at the start of simplification.

Combining Like Terms

Combining like terms is a fundamental technique to simplify algebraic expressions by reducing the number of similar terms. "Like terms" are terms that have the same variable raised to the same power.

Here’s how you can combine like terms effectively:

• Identify terms with the same variable and power, regardless of their coefficients.
• Add or subtract their coefficients, while the variable part remains unchanged.
• This helps in minimizing the expression's length and complexity.

In the expression after removing parentheses, you see like terms such as \(3x\) and \(2x\). Combining them gives you \(x\), effectively simplifying your expression. Every step should bring you closer to fewer and simpler terms.

Distribution

Distribution is the process of applying a factor across terms inside parentheses, a step required when simplifying through the removal of grouping symbols. It involves multiplying every term inside the parentheses by the term outside.

To distribute properly:

• Identify the factor outside the parentheses.
• Multiply this factor by each term inside the parentheses, adjusting any signs accordingly.
• This step prepares the expression for further simplification by flattening it.

In the example \((7x - [3x - (2x - 4) + 2] - x)\), distributing the negative sign as a factor eliminated the grouping.Applying distribution wisely ensures smooth transformation of complex expressions into manageable forms.

Algebraic Simplification

Algebraic simplification involves several processes, all aimed at making expressions easier to work with. This is achieved by applying the techniques of parentheses removal, distribution, and combining like terms.

Steps for successful algebraic simplification:

• First, carefully remove all parentheses by resolving operations or applying distribution.
• Next, identify and combine like terms to consolidate the expression.
• Finally, review the expression, checking arithmetic and ensuring no further combinations are possible.

In our original example, the simplification process results in:\[5x - 6\]Such expressions are easier to interpret and solve, allowing for clear insight into the problem. Mastering simplification ensures precise and swift solutions to algebraic problems.