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Chapter 1: Problem 86

Write an equivalent expression without using grouping symbols. $$ -(3 x+5) $$

Short Answer

Expert verified
-3x - 5

Step by step solution

01

Distribute the Negative Sign

To remove the parentheses, distribute the negative sign across each term inside the parentheses. This means you'll multiply each term by -1.
02

Multiply Each Term by -1

First, multiply \(3x\) by \(-1\) which gives \(-3x\). Then, multiply \(5\) by \(-1\) which gives \(-5\).
03

Combine Terms

Combine the results from the previous step to get the final, simplified expression without any grouping symbols. The final expression is \(-3x - 5\).

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Key Concepts

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

Distributing Negative Sign
Distributing a negative sign involves multiplying each term within the parentheses by -1. This is a crucial step to simplify expressions, especially when negative signs are involved.

In our exercise:
  • The expression inside the parentheses is \( 3x + 5 \).
  • We distribute the negative sign to both \( 3x \) and \( 5 \).
  • Mathematically, we perform \((-1) \times 3x \) and \((-1) \times 5 \).
Parentheses Removal
Removing parentheses is important for simplifying expressions. In many mathematical expressions, parentheses group terms and operations together. To simplify an expression, you need to 'break open' the group.

After distributing the negative sign, we essentially remove the parentheses by treating each term individually.

For this exercise:
  • Start with the expression \( -(3x + 5) \).
  • Distribute the negative sign resulting in \( -3x - 5 \).
  • Now, the expression no longer has parentheses and is fully simplified.
Multiplying Terms
Understanding how to multiply terms is fundamental when simplifying expressions.

Multiplying terms by -1 essentially changes their sign:
  • For \( 3x \), multiplying by -1 gives \( -3x \).
  • For 5, multiplying by -1 gives \( -5 \).
Let's break down the multiplication in our example:

  • \( -1 \times 3x = -3x \)
  • \( -1 \times 5 = -5 \)
Combining these, we get the simplified expression \( -3x - 5 \).

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