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Chapter 1: Problem 83

Simplify each expression. $$ (-3 x+4)-(6 x-6) $$

Short Answer

Expert verified
-9x + 10

Step by step solution

01

Distribute the Negative Sign

Distribute the negative sign through the second expression by multiplying each term inside the parentheses by -1. This results in (-3x + 4) - 6x + 6.
02

Remove Parentheses

Now we can remove the parentheses: -3x + 4 - 6x + 6.
03

Combine Like Terms

Combine the x terms and the constant terms separately. -3x - 6x = -9x for the x terms, and 4 + 6 = 10 for the constants. This gives us -9x + 10.

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Key Concepts

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

Distributing the Negative Sign
When simplifying algebraic expressions, one key step is distributing the negative sign. This can be tricky but is crucial for getting the right answer. When you see a negative sign before a parenthesis, you need to multiply everything inside the parenthesis by -1. For example, if you have (-3x + 4) - (6x - 6), you'll multiply both \(6x\) and \(-6\) by -1. This results in: (-3x + 4) - 6x + 6.

By doing this, the terms inside the second set of parentheses are effectively flipped: positive becomes negative, and negative becomes positive. This step ensures that all terms have the correct sign before you start combining them.
Removing Parentheses
Once you've distributed any negative signs within the expression, the next step is to remove the parentheses. This makes it easier to simplify the expression. In our example, after distributing the negative sign we got (-3x + 4) - 6x + 6.

Now, we simply remove the parentheses:
-3x + 4 - 6x + 6.

Removing parentheses is straightforward once you've ensured that any necessary distribution has been completed. It helps to look at each term separately, paying attention to their signs.
Combining Like Terms
After removing the parentheses, it's time to combine like terms. Like terms are those that have the same variable raised to the same power. For example, \(-3x\) and \(-6x\) are like terms because they both have \(x\) to the first power. Similarly, 4 and 6 are like terms as they are both constants.

In our problem, -3x + 4 - 6x + 6:
  • Combine the \(x\) terms: -3x - 6x = -9x.
  • Combine the constant terms: 4 + 6 = 10.
This gives us the final simplified expression: -9x + 10.

By systematically combining like terms, you ensure the expression is as simple as possible.

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Most popular questions from this chapter

Determine whether each statement is true or false. Explain your answer. $$\begin{aligned}&A=\\{1,3,5,7,9\\} \quad B=\\{2,4,6,8\\}\\\&C=\\{1,2,3,4,5\\} \quad N=\\{1,2,3, \dots\\}\end{aligned}$$ $$A \subseteq N$$

Combine like terms. $$ -3 a+15 a $$

Perform these computations. $$55-0.1$$

Evaluate each radical. $$\sqrt{49}$$

Perform these computations. $$64-|-12|$$
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