Problem 87 - Algebra A Combined Function

Question

Simplify each expression. See Section \(1.8 .\) \(-(x-1)+x\)

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Identify the Expression Terms

The expression given is \(-(x-1)+x\). We need to simplify this expression by first identifying and understanding the terms included within the expression.

Step 2: Apply the Negative Sign

Distribute the negative sign across the terms inside the parentheses: \(-(x-1) = -x + 1\).

Step 3: Rewrite the Expression

Rewrite the expression with the expanded form: \(-x + 1 + x\).

Step 4: Combine Like Terms

Identify and combine like terms: \(-x + x + 1\). Both \-x\ and \x\ are like terms and combine to \0\, leaving us with just \1\.

Key Concepts

Combine Like Terms

One of the foundational skills in algebra is the ability to combine like terms effectively. Let's define what like terms are. These are terms that have the same variable raised to the same power. For example, in the expression \ \(-x + x + 1\), the terms \(-x\) and \(x\) are considered like terms because they both contain the variable \(x\) raised to the power of 1. Although they have opposite signs, their coefficients can be directly added or subtracted. When combining like terms, always pay attention to the coefficients—the numerical part of each term. In this example, the coefficients are \(-1\) and \(1\), and when combined, they result in \(0\), simplifying the expression to \(1\).
Here are some tips for combining like terms:

• Identify terms with the same variables and exponents.
• Add or subtract the coefficients of these like terms.
• Keep track of positive and negative signs to avoid errors.

Distributive Property

The distributive property is a key principle when simplifying expressions, especially ones containing parentheses. This property states that for any numbers \(a\), \(b\), and \(c\), the equation \(a(b + c) = ab + ac\) holds true. Applying this property allows you to multiply a single term by each term inside the parentheses separately. For instance, in the original exercise \ \(-(x-1)+x\), the distributive property is not directly applied because there's a negative sign instead of a numerical coefficient that multiplies the terms inside the parentheses Despite that, understanding the distributive nature is still essential in different parts of algebraic manipulation.

• Apply multiplication to each term inside the parentheses.
• Resulting terms should be written out separately after distribution.
• Always simplify further by combining like terms if possible.

While the exercise skips directly to handling the negative sign, grasping the distributive rule will aid in more complex problems in future studies.

Negative Sign Distribution

Understanding how to distribute a negative sign in algebra is crucial for correctly simplifying expressions. In an expression like \ \(-(x-1)+x\), the negative sign affects every term inside the parentheses. This means that when you see a negative sign followed by parentheses, it's equivalent to multiplying every term inside by \(-1\). Let's take a closer look at how this works:

• The expression \(-(x-1)\) can be rewritten by distributing the negative sign as \(-x + 1\). Here, \(-1\) times \(x\) becomes \(-x\), and \(-1\) times \(-1\) becomes \(+1\).
• After distributing, simplify the expression by combining like terms, if any.

This understanding is particularly useful when dealing with complex expressions, ensuring that you maintain the correct operations throughout. Always remember to take the sign with the number when performing distribution. This simple rule helps prevent mistakes especially when multiple terms are involved.