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Chapter 8: Problem 88

Simplify: \(3\left(\frac{x-2}{3}\right)+2\)

Short Answer

Expert verified
The simplified form of the expression is \(x\).

Step by step solution

01

Distribute the Multiplication

Distribute the 3 across the terms inside the parentheses: \(3 * \frac{x}{3}\) and \(3 * \frac{-2}{3}\). This gives \(x - 2\).
02

Combine Like Terms

Add the constant term 2 to the -2 obtained from step 1. This gives \(x - 2 + 2\).
03

Simplify

Simplify \(x - 2 + 2\) to get \(x\).

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Key Concepts

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

Distributive Property
The distributive property is a fundamental concept in algebra crucial for simplifying expressions. It involves multiplying a single term by each term within parentheses. In our exercise,
  • we use the distributive property to multiply 3 by both terms inside the parentheses: \( \frac{x-2}{3} \).
  • This results in two separate operations: \( 3 * \frac{x}{3} \) and \( 3 * \frac{-2}{3} \).
  • The first operation, \( 3 * \frac{x}{3} \), simplifies to \( x \).
  • The second operation, \( 3 * \frac{-2}{3} \), simplifies to \( -2 \).
Mastering the distributive property helps in efficiently breaking down and solving more complex algebraic expressions.
Combining Like Terms
Combining like terms is another essential skill in algebra that simplifies expressions to their simplest form by merging terms that have the same variables and exponents. In the solution exercise:
  • After using the distributive property, we arrived at \( x - 2 \).
  • Adding the constant term 2 from the original expression results in \( x - 2 + 2 \).
  • The like terms \( -2 \) and \( +2 \) are combined to simplify the expression to \( x \).
This process streamlines expressions and makes further calculations more manageable. Remember, only terms that have identical variable parts can be combined.
Rational Expressions
A rational expression is a fraction where both the numerator and the denominator are polynomials. Though our example is simple, it is based on understanding rational expressions:
  • We began with \( \frac{x-2}{3} \), which is a rational expression.
  • Rational expressions require careful handling, especially when simplifying using operations like the distributive property.
  • Ensure you simplify both the numerator and denominator appropriately before carrying out further operations.
Understanding rational expressions aids in tackling more complex algebraic equations and fractions easily.
Basic Algebra Concepts
In algebra, understanding the basic concepts like variables, constants, and operations is crucial. In the exercise, the variables and constants are important elements:
  • We handled \( x \) as a variable, representing an unknown quantity.
  • The numbers \( 2 \) and \( -2 \) are constant terms that needed to be combined.
  • Basic operations such as multiplication and addition played key roles in simplifying the expression.
Grasping these foundational concepts empowers you to tackle algebra with confidence and solve even more challenging problems.

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