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Chapter 2: Problem 58

Simplify each expression. See Examples 3 through 7 . $$ 8+4(3 x-4) $$

Short Answer

Expert verified
The simplified expression is \( 12x - 8 \).

Step by step solution

01

Distribute the Multiplier inside Parentheses

To start simplifying the expression, apply the distributive property to multiply the number 4 with each term inside the parentheses (3x - 4). This gives: \( 4 \times 3x \) and \( 4 \times (-4) \). This results in new terms: \( 12x \) and \( -16 \).
02

Rewrite the Expression with the Distributed Terms

Substitute the distributed terms back into the original expression, replacing the parenthetical expression: \( 8 + 12x - 16 \).
03

Combine Like Terms

Now, identify and combine like terms to simplify further. The terms \( 8 \) and \( -16 \) are constant terms that can be combined: \( 8 - 16 = -8 \). Thus, the expression becomes \( 12x - 8 \).

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Key Concepts

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

Understanding the Distributive Property
In algebra, the distributive property is a useful technique that can simplify expressions. This property helps us to multiply a single term by each term inside a set of parentheses. For the expression given, 4 is multiplied with each term inside the parentheses: 3x and -4. The expression \( 4(3x - 4) \) becomes \( 4 \times 3x + 4 \times (-4) \). This simplifies to two new terms:\[ 12x - 16. \]
Applying the distributive property correctly depends on treating both the constants and variables in the expression properly.
  • If you distribute right, each term inside the parentheses receives the multiplier equally.
  • Watch out for negative signs, as they affect the sign of the product.
Once you handle these aspects, you can simplify expressions with ease, breaking them down into manageable parts.
Simplifying Algebraic Expressions
Simplifying expressions is a key step towards solving algebra problems. Once you distribute terms as shown in the previous section, the expression \( 8 + 12x - 16 \) is ready for simplification.
This involves reorganizing and adjusting terms so that the expression is in its simplest form.
In this example, we need to clean up by removing parentheses and simplifying what鈥檚 left by combining terms,
making it easier to evaluate or solve equations that include this expression.
  • After distribution, remove parentheses and look for opportunities to clean up the expression.
  • Check if any terms can be reorganized or simplified, such as switching the order of like terms.
  • This practice provides a clearer expression and helps in solving more complex algebraic equations.
Combining Like Terms Effectively
Combining like terms is crucial for making expressions easier to manage. In our expression \( 8 + 12x - 16 \), the like terms \( 8 \) and \(-16\) are constants that can be combined to streamline the workflow.
This process involves adding or subtracting constants and coefficients of similar variables.
The expression thus becomes \( 12x - 8 \) as a result of combining \( 8 \) and \( -16 \).
  • Identify terms with the same variable component, as they can be added or subtracted together.
  • Constant terms (like numbers without variables) are also like terms and can be combined easily.
  • After combining, the expression becomes much simpler, making further manipulations and evaluations straightforward.

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Perform the indicated operations. See Sections 1.5 and 1.6 $$ 3+(-7) $$
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