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Chapter 2: Problem 44

Simplify each expression. See Examples 3 through 7 . $$ -3(2 x+5)-6 x $$

Short Answer

Expert verified
The simplified expression is \(-12x - 15\).

Step by step solution

01

Distribute -3

First, distribute the \(-3\) to both terms inside the parentheses \((2x + 5)\). This means multiplying \(-3\) by \(2x\) and then by \(5\):\[-3(2x) + (-3)(5) = -6x - 15\]
02

Combine Like Terms

Now, combine the like terms in the expression \(-6x - 15 - 6x\). Combine \(-6x\) and \(-6x\):\[-6x - 6x = -12x\]The expression simplifies to \(-12x - 15\).

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Key Concepts

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

Distributive Property
When encountering an expression with parentheses, like \(-3(2x+5)\), you can use the distributive property to help simplify it. The distributive property allows you to multiply a single term by each term inside the parentheses separately.
Think of it like sharing a task equally among friends.
  • In our example, we need to multiply \(-3\) with each part of \(2x + 5\).
  • First, \(-3\) times \(2x\) gives us \(-6x\).
  • Next, \(-3\) times \(5\) gives us \(-15\).
So, by distributing, you transform \(-3(2x+5)\) into \(-6x - 15\). Remember to pay attention to signs (positive and negative) while distributing them. Each part must have its sign correctly applied.
Combining Like Terms
Once you have simplified using the distributive property, the next step is combining like terms. This is crucial to further simplify an expression. Like terms are terms in an expression that have the same variable raised to the same power.
In our example, the expression after distributing is \(-6x - 15 - 6x\).
  • You can combine the terms \(-6x\) and \(-6x\) because they both contain \(x\).
  • Adding \(-6x\) and \(-6x\) results in \(-12x\).
The \(-15\) has no like terms to combine with, so it remains as it is.
By combining like terms correctly, the expression simplifies nicely to \(-12x - 15\).
Linear Expressions
Linear expressions are algebraic expressions where each term is either a constant or the product of a constant and a single variable. In other words, they do not involve exponents higher than one.
In our final simplified expression \(-12x - 15\), we maintain a linear form.
  • Each term involves the variable \(x\) or is a constant without any exponents.
  • No parts of the expression have squared, cubed, or any other higher powers of \(x\).
This linear nature makes such expressions easier to handle and further points to their balance and symmetry.
Understanding linear expressions is essential, as they form the basis of equations that model real-world scenarios, making them fundamental in algebra education.

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