Problem 57 - Algebra for College Students

Question

Simplify by removing the inner parentheses first and working outward. $$3 x-[5 x-(x+6)]$$

Step-by-Step Solution

Answer

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

Step 1: Remove the Innermost Parentheses

Start by simplifying the expression inside the innermost parentheses: \(x + 6\). Since there is no operation before the parentheses, the expression remains as it is.

Step 2: Eliminate Second Layer of Parentheses

The expression contains \(5x - (x + 6)\). Distribute the minus sign across the terms inside the parentheses, resulting in \(5x - x - 6\). Simplify this to get \(4x - 6\).

Step 3: Substitute Back into the Expression

Substitute \(4x - 6\) back into the original expression, replacing \(5x - (x + 6)\) with \(4x - 6\). This results in \(3x - [4x - 6]\).

Step 4: Remove Outer Parentheses

Similar to step 2, remove the outermost parentheses by distributing the negative sign: \(3x - 4x + 6\).

Step 5: Combine Like Terms

Combine the terms to simplify the expression: \(3x - 4x + 6 = -x + 6\).

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle that helps you simplify expressions. It allows you to multiply a single term by all terms within parentheses. For example, if you have parentheses preceded by a negative sign, like in the expression \( -(x+6) \), you apply the distributive property by multiplying \(-1\) with each term:

• \( -1 \times x = -x \)
• \( -1 \times 6 = -6 \)

This results in \( -x - 6 \). By using the distributive property, you can remove parentheses and simplify the expressions easier. This step is crucial in our original exercise, where we distributed the minus sign across \((x + 6)\) to simplify the part of the equation within the brackets.

Combining Like Terms

Combining like terms is a technique used in algebra to simplify expressions by grouping and combining terms that have the same variables raised to the same power. For instance, in the expression \(3x - 4x + 6\), the terms \(3x\) and \(-4x\) are like terms because they both include the variable \(x\).
You combine them by simply adding their coefficients:

• \(3 - 4 = -1\)

This yields \(-x\). The constant \(6\) has no like terms here, so it remains unchanged.
Combining like terms makes expressions neater and easier to evaluate. This final combination of terms was the last step in our solution, resulting in the simplified expression \(-x + 6\).
Being comfortable with this operation is essential for tackling complex algebraic expressions.

Parentheses in Algebra

Parentheses are used in algebra to indicate which operations should be performed first and to group terms and expressions logically. In equations and expressions, the rules of parentheses influence the order of operations.
Here's why they're important:

• They ensure certain calculations are done first according to the order of operations.
• They help neatly organize parts of an expression.

In our original exercise, we first solved the expressions inside the innermost parentheses \((x + 6)\). Although simple here, sometimes it dictates which terms are subjected to particular operations first, such as multiplication or distribution.
Once you properly handle parentheses by using the distributive property, the rest of the expression typically simplifies further. Mastering the handling of parentheses is key to correctly simplifying algebraic expressions.