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Chapter 7: Problem 137

In the following exercises, simplify using the distributive property. $$4(6 x-1)-(-8)$$

Short Answer

Expert verified
The simplified expression is \(24x + 4\).

Step by step solution

01

Apply the Distributive Property

Distribute the 4 into each term inside the parentheses. The distributive property states that for any numbers a, b, and c, the expression a(b + c) can be expanded to ab + ac. In this case, we have: \[4(6x - 1) = 4 \times 6x - 4 \times 1 = 24x - 4\]
02

Simplify the Expression

Now, simplify the second part of the expression by removing the negative sign. We have: \[24x - 4 - (-8)\] which is equivalent to \[24x - 4 + 8\]
03

Combine Like Terms

Combine the constant terms in the expression. Add -4 and 8 together: \[24x - 4 + 8 = 24x + 4\]

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Key Concepts

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

simplify expressions
When simplifying expressions, the goal is to make them as easy to manage as possible. Here are a few key steps:

Start by using properties like the distributive property, which lets you multiply a single term across a binomial or polynomial. For example, in the expression \(4(6x - 1) - (-8)\), we distribute the 4 across (6x - 1) to get 24x - 4.

Remain consistent and orderly while simplifying. Always perform operations step by step, ensuring every term is addressed. This discipline will help avoid errors and make the end result cleaner.
like terms
Identifying and combining like terms is an essential part of simplifying expressions. Like terms are terms in an expression that have the same variable raised to the same power. In the expression \(24x - 4 + 8\), we have:

  • One term with the variable x: \(24x\)
  • Two constant terms: \(-4\) and \(8\)

To combine like terms, simply add or subtract the coefficients. For example, add \(-4\) and \(8\) together to get \(4\). So, the expression simplifies to \(24x + 4\).
negative signs
Dealing with negative signs can be tricky but understanding a few rules can make it easier.

First, remember that subtracting a negative number is the same as adding the positive version of that number. For example, in our expression \(24x - 4 - (-8)\), the \(-(-8))\) becomes \(+8\).

Negative signs also appear when you distribute a negative multiplier across parentheses. For instance, \(-3(2x - 5)\) will become \(-6x + 15\). Each term inside gets multiplied by \(-3\).

Keeping track of negative signs throughout your calculations will help to avoid mistakes and ensure your simplified expression is accurate.

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