Problem 34 - Algebra for College Students

Question

Simplify the algebraic expressions in Problems \(15-34\) by removing parentheses and combining similar terms. $$4(-x-1)+3(-2 x-5)-2(x+1)$$

Step-by-Step Solution

Answer

The simplified expression is \(-12x - 21\).

Step 1: Distribute the First Term

Start by distributing the 4 into the first parenthesis: \[ 4(-x-1) = 4(-x) + 4(-1) = -4x - 4 \]

Step 2: Distribute the Second Term

Distribute the 3 into the second parenthesis: \[ 3(-2x-5) = 3(-2x) + 3(-5) = -6x - 15 \]

Step 3: Distribute the Third Term

Distribute the -2 into the third parenthesis:\[ -2(x+1) = -2(x) + -2(1) = -2x - 2 \]

Step 4: Combine Like Terms

Now, add all the terms from the distribution steps together:\[ -4x - 4 - 6x - 15 - 2x - 2 \] Combine the like terms (those containing \(x\) and the constants):\[ (-4x - 6x - 2x) + (-4 - 15 - 2) = -12x - 21 \]

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential skill in algebra that involves reducing complex equations or expressions to their simplest form. This process makes it easier to analyze and solve mathematical problems. The goal is to perform operations and eliminate unnecessary parentheses or like terms.
Simplifying helps in clarifying expressions for easier computation and understanding, paving the way for solving even the most complex algebraic equations effectively. It involves several steps: distributing terms, combining like terms, and managing parentheses efficiently. With practice, these skills become second nature, simplifying your algebra journey.

Distributing Terms

When you distribute terms in an expression, you apply the distributive property to multiply each term inside the parentheses by the factor outside the parentheses. For instance, in the expression \( 4(-x-1) \), you distribute 4 across each term inside the parentheses:

• \( 4(-x) \), which becomes \( -4x \)
• \( 4(-1) \), which becomes \( -4 \)

This operation breaks down complex expressions into simpler parts, making further simplification easier. The distributive property is key in algebra because it allows you to eliminate parentheses and rearrange equations into a more workable format. Remember, correct distribution lays the foundation for combining like terms accurately.

Combining Like Terms

Combining like terms involves adding or subtracting terms that have similar variables and powers. This step reduces the expression further to its simplest form. In our example, after distributing all terms, we have the expression:
\[-4x - 4 - 6x - 15 - 2x - 2\]

• The terms \((-4x, -6x, -2x)\) are like terms because they all contain the variable \(x\).
• The constant terms \((-4, -15, -2)\) are combined separately.

Thus, combining \((-4x, -6x, -2x)\) results in \(-12x\) and \((-4, -15, -2)\) simplifies to \(-21\). The final simplified expression is \(-12x - 21\). Merging like terms is crucial for clarity and ease in advancing through algebraic operations.

Parentheses in Algebra

Parentheses in algebra are used to group terms and dictate the order in which operations should be performed. They indicate which parts of an expression should be addressed first, following the order of operations (PEMDAS/BODMAS). In our particular problem, parentheses determine how terms are distributed:

• They help structure expressions for sequential solving. For instance, \( 4(-x-1) \) ensures that both terms inside are multiplied by 4.
• Parentheses prevent miscalculations and maintain the intended equation order.

Avoidance of errors when handling complex equations often hinges on proper attention to parentheses. By selectively distributing over these grouped terms, you simplify the expression step by step effectively, ensuring accuracy in your final result.