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Chapter 5: Problem 109

Simplify. \(3 x+7-(x+3)\)

Short Answer

Expert verified
The simplified expression is \(2x + 4\).

Step by step solution

01

- Distribute the Negative Sign

First, distribute the negative sign inside the parentheses. This means changing the sign of each term inside the parentheses.Original expression: \(3x + 7 - (x + 3)\)After distributing the negative sign: \(3x + 7 - x - 3\)
02

- Combine Like Terms

Next, combine like terms. Like terms are terms that contain the same variable raised to the same power.Combine the \(x\)-terms:\(3x - x = 2x\)Combine the constant terms:\(7 - 3 = 4\)
03

- Write the Simplified Expression

After combining like terms, write down the simplified expression.Simplified expression: \(2x + 4\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combining Like Terms
Combining like terms is a crucial step in simplifying algebraic expressions. It involves adding or subtracting terms that have the same variables raised to the same power. For instance, in the expression provided,
  • We identify the like terms: the terms involving x (which are 3x and -x) and the constant terms (which are 7 and -3).
  • After identifying, we perform the operations:
    • For x terms: 3x - x = 2x.
    • For constant terms: 7 - 3 = 4.
This makes the expression simpler and easier to manage. Always look for terms with the same variables and combine them to reduce the expression.
Distributive Property
The distributive property is an essential tool in algebra that lets us open up parentheses and simplify expressions. It states that a(b + c) = ab + ac. In this problem, though,we specifically address distributing a negative sign, which turns every term inside the parentheses into its opposite. Here鈥檚 how this works step-by-step:
  • Start with the expression: 3x + 7 - (x + 3).
  • Distribute the negative sign through the parentheses:
    • -(x) = -x
    • -(3) = -3
    This changes the expression to: 3x + 7 - x - 3.
By carefully applying this property, we ensure all signs are correctly adjusted, which makes the next steps of simplifying much smoother.
Elementary Algebra
Elementary algebra forms the foundation for understanding more complex mathematical concepts. Here are a few key points to remember:
  • Algebra uses symbols, usually letters like x and y, to stand for numbers.
  • It involves operations such as addition, subtraction, multiplication, and division, along with following specific rules such as the distributive property or combining like terms.
  • The goal is to simplify expressions or solve equations by isolating variables.
In the provided exercise, simplifying the expression 3x + 7 - (x + 3) involves both the distributive property and combining like terms. By breaking down the steps methodically, we end up with a much simpler expression: 2x + 4. This showcases the power of elementary algebra in making complex problems more manageable.

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