Problem 91 - Intermediate Algebra

Question

Simplify each expression. $$-(a+2 A+1)-(a-A+2)$$

Step-by-Step Solution

Answer

The simplified expression is \(-2a - A - 3\).

Step 1: Distribute the Negative Sign

Begin by distributing the negative sign across the terms inside each parenthesis. The expression becomes: \[-(a + 2A + 1) - (a - A + 2) = -a - 2A - 1 - a + A - 2\]

Step 2: Combine Like Terms

Now, combine the like terms. The expression becomes: \[-a - a - 2A + A - 1 - 2\]. This results in: \[-2a - A - 3\].

Key Concepts

Distributive Property

The Distributive Property is a useful tool in algebra that helps in simplifying expressions, especially those with parentheses. Generally, the property states that any number multiplied by a sum or difference inside parentheses can be distributed, or "spread out," as a multiplication across each term within the parentheses. For example, if you have a term like \(x(a + b)\), it simplifies to \(xa + xb\).
This rule is crucial when dealing with complex expressions because it enables you to expand them into simpler forms where you can more easily identify like terms. In the given exercise, you applied the distributive property when you saw the negative sign before two sets of parentheses.
- You changed \(- (a + 2A + 1)\) to \(-a - 2A - 1\).Similarly, the other part changed from \(- (a - A + 2)\) to \(-a + A - 2\).

The distributive property also helps in factoring and on your journey to mastering algebraic expressions.

Combining Like Terms

Combining like terms is another essential skill in algebra, and it comes into play once you've expanded your expression using the distributive property. Like terms are terms in an expression that have the exact same variable raised to the same power. For instance, \(3a\) and \(-2a\) are like terms, while \(3a\) and \(3b\) are not because they contain different variables.

In the problem we analyzed, after distributing the negative sign, we reached an expression: \(-a - 2A - 1 - a + A - 2\).
The process of combining involves adding or subtracting the coefficients of like terms:

• For \(-a - a\), you get \(-2a\).
• For \(-2A + A\), you resolve it to \(-A\).
• For the constants: \(-1 - 2\) becomes \(-3\).

By combining terms this way, you manage to condense the complex expression into a simpler one: \(-2a - A - 3\).

Combining like terms is not only about making an expression shorter; it also clarifies the relationships between variables and constants, which is crucial for solving equations.

Negative Sign Distribution

Understanding how to handle negative signs in algebra is vital, especially when they appear outside parentheses. Often, this involves distributing a negative sign across a group of terms inside the parentheses, effectively switching the signs of each term.

In the exercise at hand, the expression is \(- (a + 2A + 1) - (a - A + 2)\).When you distribute the negative sign, you need to think about multiplying each term inside the parentheses by \(-1\). This means \( a\) becomes \(-a\), \(2A\) becomes \(-2A\), and so forth.

• The expression \(- (a + 2A + 1)\) transforms to \(-a - 2A - 1\).
• Similarly, \(- (a - A + 2)\) becomes \(-a + A - 2\).

This technique is fundamental when changing the form of expressions, specifically when aiming to simplify or solve equations. Missteps with handling negative signs can result in incorrect solutions, so accuracy here is key to mastering algebra.