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Chapter 7: Problem 58

State the restrictions and then simplify. $$ 3 x+11+3 x $$

Short Answer

Expert verified
The simplified expression is \( 6x + 11 \) with no restrictions on \( x \).

Step by step solution

01

Identify Like Terms

The expression given is \( 3x + 11 + 3x \). First, identify terms of the same kind that can be combined. Here, \( 3x \) and \( 3x \) are like terms because they both contain the variable \( x \) with the same exponent.
02

Combine Like Terms

Add the coefficients of the like terms identified. Combine \( 3x + 3x \) to get \( 6x \). The expression then becomes \( 6x + 11 \).
03

State the Restrictions

Since we are not dividing by any variables or having variables in a denominator, there are no restrictions on the variable \( x \). However, if there was division involving \( x \) that introduced a denominator, \( x \) could not be zero.
04

Final Simplified Expression

The simplified form of the original expression is \( 6x + 11 \). There are no additional simplifications possible, as \( 6x \) and \( 11 \) are unlike terms.

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Key Concepts

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

Like Terms
In algebra, "like terms" are terms that have the exact same variable part. This means they must have the same variables raised to the same powers. For instance, in the expression \(3x + 11 + 3x\), the terms \(3x\) and \(3x\) are like terms because both contain the variable \(x\).
Recognizing like terms is crucial because it allows you to combine them, simplifying the expression.
  • Like terms can be combined by adding or subtracting their coefficients.
  • Unlike terms, such as \(6x\) and \(11\) in \(6x + 11\), cannot be combined.
Combining like terms can make solving algebraic expressions much easier and is a foundational skill in algebra.
Simplifying Expressions
Simplifying expressions is all about making them look cleaner and easier to work with. It involves combining like terms and reducing expressions as much as possible without changing their value.
Take the expression \(3x + 11 + 3x\) as an example. To simplify:
  • Find like terms: The terms \(3x\) and \(3x\) are like terms.
  • Combine the coefficients of like terms: Adding the coefficients of \(3x\) gives us \(3 + 3 = 6\), so you get \(6x\).
  • Write down the simplified expression: \(6x + 11\).
Remember, simplifying can be as simple as tidying up expression components. It helps in effectively solving equations and prepares the expression for further algebraic operations.
Variable Restrictions
Variables in algebraic expressions can sometimes come with restrictions. These are conditions that limit the values a variable can take.
In the expression \(3x + 11 + 3x\), there are no restrictions because:
  • There is no division by the variable \(x\) involved.
  • No square roots or other operations that might restrict \(x\).
However, if an expression has a denominator like \(\frac{1}{x}\), then \(x\) cannot be zero, as division by zero is undefined.
Understanding variable restrictions helps avoid undefined values or impossible outcomes in algebraic operations. Always consider these restrictions when handling expressions with variables.

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