Problem 58 - Algebra for College Students

Question

Simplify by removing the inner parentheses first and working outward. $$7 x-[2 x-(-x-4)]$$

Step-by-Step Solution

Answer

The simplified expression is \(6x + 4\).

Step 1: Distribute the Negative Sign Inside the Parentheses

Start by looking at the inner parentheses expression \((-x - 4)\). The negative sign in front of the parentheses \(-(-x - 4)\) means you need to distribute it. This will change the signs of the terms inside the parentheses: \[ -(-x - 4) = x + 4. \]

Step 2: Simplify the Expression Inside the Brackets

Now substitute the result from Step 1 back into the expression:\[ 2x - (x + 4) \].Since there is a subtraction sign in front of the \( x + 4 \), distribute it:\[ 2x - x - 4 \].Simplify by combining like terms:\[ x - 4 \].

Step 3: Substitute Back and Simplify the Entire Expression

Replace \(2x - (-x-4)\) with its simplified form \(x - 4\) in the original expression:\[ 7x - (x - 4) \].Distribute the negative sign:\[ 7x - x + 4 \].Combine like terms:\[ 6x + 4 \].

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to simplify expressions, particularly when dealing with parentheses. It states that multiplying a number by a sum is the same as doing the sum of the individual products. Mathematically, it is represented as:\[ a(b + c) = ab + ac \]To apply the distributive property effectively, you should:

• Identify the term outside the parentheses that will be used for distribution.
• Multiply this term by each term inside the parentheses.
• Be mindful of the signs (positive and negative) of each term.

For example, applying it in the given exercise, we see it used to change the signs inside the parentheses \(-(-x - 4)\) by multiplying each term by \(-1\). This results in \(x + 4\).
Likewise, when redistributing in \(2x - (x + 4)\), we distribute the negative sign \(-\) across the terms \(x + 4\) to simplify the expression further.
This process highlights how the distributive property helps in removing parentheses and flattening expressions for easier manipulation.

Combining Like Terms

Combining like terms is a process used in algebra to simplify expressions. Like terms are terms that have the same variables raised to the same power. Essentially, these terms can be added or subtracted from each other.
When simplifying expressions, look for terms that have the same variable and exponent.
For instance, if you have terms such as \(3x\), \(4x\), and \(-x\), these can be combined because they all contain the variable \(x\).

• Align the terms such that similar variables appear together. This often helps in visualizing the computation process better.
• Perform arithmetic operations based on their coefficients while keeping the variable part intact.

In the solution provided, the final step involves combining the like terms of \(7x - x\), which simplifies to \(6x\) by subtracting the coefficient \(1\) from \(7\).
This step simplifies the calculation significantly, leading to an easy-to-interpret expression \(6x + 4\).
Make sure to always double-check for any terms that might be overlooked, as missing a term can lead to incorrect results.

Parentheses in Algebra

In algebra, parentheses are used to determine the order in which operations should be performed. This is crucial as it directly influences the outcome of calculations.
Parentheses create a group of terms that should be considered together, acting as an encapsulation to dictate priority.

• Start simplifying expressions from the innermost parentheses outward to ensure proper handling.
• Once an operation inside parentheses is completed, rewrite the expression to reflect those changes.
• Be attentive to operations before and after parentheses, often requiring the use of the distributive property or sign changes.

For the given algebraic expression \(7x - [2x - (-x - 4)]\), removing the innermost parentheses first was essential. The process continued outward, ensuring an accurate simplification order.
After performing operations on \(-(-x - 4)\), parentheses were further cleared in \(2x - (x + 4)\), eventually leading to the simplified whole expression.
Remember, meticulous attention to parentheses and the order of operations can prevent errors and aid in reaching the correct solution efficiently.