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Chapter 3: Problem 65

Simplify each expression. $$ -(4 x-10)+2(3 x+5) $$

Short Answer

Expert verified
The expression simplifies to \(2x + 20\).

Step by step solution

01

Distribute the Minus Sign

Apply the minus sign to both terms inside the parenthesis: \[-(4x - 10) = -4x + 10\]
02

Distribute the 2

Multiply 2 by both terms inside the parenthesis: \[2(3x + 5) = 6x + 10\]
03

Combine the Expressions

Combine the expressions from Step 1 and Step 2: \[-4x + 10 + 6x + 10\]
04

Combine Like Terms

Add the like terms (terms with x and constant terms): \[-4x + 6x = 2x\] and \[10 + 10 = 20\]The simplified form of the expression is: \[2x + 20\]

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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 useful tool in algebra. It allows you to eliminate parentheses by distributing a number across terms inside it. Here's how it works: When you have an expression like \( a(b + c) \), you distribute the \(a\) to both \(b\) and \(c\), resulting in \( ab + ac \). This rule helps in breaking down more complex expressions into simpler ones. In the given exercise, the Distributive Property was applied twice:
  • First, by distributing the minus sign across the terms within the parenthesis: \[-(4x - 10) = -4x + 10\]
  • Second, by distributing the \(2\) across each term in the parenthesis: \[2(3x + 5) = 6x + 10\]
Understanding this property well can make it easier to manipulate and solve equations in algebra.
Like Terms
Like terms are terms that have the exact same variable part. Only the coefficients can be different. To simplify an expression, you should combine like terms. It鈥檚 essential to identify which terms in an algebraic expression are alike. For example, in the expression \(-4x + 10 + 6x + 10\), the terms \(-4x\) and \(6x\) are like terms. Similarly, the numbers \(10\) and \(10\) are both constant terms and should be combined.Combining like terms means adding the coefficients of the terms:
  • For \(-4x\) and \(6x\), you add: \[-4x + 6x = 2x\]
  • For the constants \(10\) and \(10\), you add: \[10 + 10 = 20\]
Once like terms are combined, the expression becomes simpler and easier to work with.
Simplifying Expressions
Simplifying an expression means to make it as concise and readable as possible by performing all possible arithmetic operations, including distribution and combining like terms. In the given problem, the initial expression was complex due to the parentheses: \[-(4x-10)+2(3x+5)\]By applying the distributive property and combining like terms, we simplified it to:\[2x + 20\] Here鈥檚 why simplifying is important:
  • It makes expressions easier to understand and work with in further calculations or algebraic operations.
  • It helps in solving equations by reducing complexity.
Remember, the key to simplifying is to systematically distribute and then combine like terms. It makes the expressions orderly and manageable, an essential skill in algebra.

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