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Chapter 6: Problem 55

Perform the indicated operations. When possible write down only the answer. $$(a-b) \div(-1)$$

Short Answer

Expert verified
-a + b

Step by step solution

01

Understand the Operation

Identify the operation to be performed. Here, we need to divide the expression \(a - b\) by -1.
02

Apply Division by -1

When dividing any number by -1, the result is the negation of that number. Therefore, divide \(a - b\) by -1: \((a - b) \div (-1) = - (a - b)\).
03

Simplify the Result

Distribute the negative sign inside the parenthesis. This yields: \(- (a - b) = -a + b\).
04

Write Down the Answer

The final simplified form of the expression is: \(-a + b\).

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Key Concepts

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

Division
In algebra, division is one of the fundamental operations that you will encounter. In this exercise, we are dealing with the division of an expression by -1. Understanding how to handle such operations is crucial for solving more complex equations later on.
When dividing by -1, you are essentially flipping the sign of the number or expression you are dividing. For instance, \[-1 \div -1 = 1\] and \[-3 \div -1 = 3\]. In our exercise, we divide the expression \(a - b\) by -1, resulting in the negation of the entire expression.

Important points to remember:
  • Dividing by -1 flips the sign of the expression.
  • Keep track of each term's sign to avoid mistakes.

Make sure to perform each step carefully and check your work often, as sign errors can lead to incorrect results.
Negative Numbers
Negative numbers play a significant role in algebra. They represent values less than zero and follow specific rules, especially when involved in operations like addition, subtraction, multiplication, and division. Understanding the behavior of negative numbers is necessary to avoid miscalculations.

Key rules to remember:
  • Dividing a positive number by -1 changes it to negative.
  • Dividing a negative number by -1 changes it to positive.
  • When you multiply or divide two negative numbers, you get a positive result.
  • When you multiply or divide a negative number with a positive number, the result is negative.

Always pay close attention to the signs when dealing with negative numbers to ensure you perform the operation correctly. In our exercise, dividing \(a - b\) by -1 led us to \(-(a - b)\). This step negated the entire expression inside the parentheses.
Simplification
Simplifying expressions is a critical aspect of algebra. It makes your final answer cleaner and easier to understand. Let's break down the simplification step in our exercise.
First, note that our expression after the division was \(-(a - b)\). Simplification involves distributing the negative sign inside the parentheses. Here's the step-by-step breakdown:
  • Negate each term inside the parenthesis: \-a becomes a\ and \-b becomes b\.
  • Combine the terms: \(-a + b\).

By distributing the negative sign correctly, we simplified our expression from \(-(a - b)\) to \(-a + b\). Simplification helps in finalizing the answer and ensures that it's in its most reduced form, making it easier to interpret and use in further calculations.

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Most popular questions from this chapter

Simplify each complex fraction. $$\frac{\frac{3}{x^{2}-1}-\frac{x-2}{x^{3}-1}}{\frac{3}{x^{2}+x+1}+\frac{x-3}{x^{3}-1}}$$

Simplify each complex fraction. Use either method. $$\frac{\frac{1}{2}-\frac{1}{4}}{\frac{1}{6}-\frac{1}{8}}$$

Discussion. Use synthetic division to find the quotient when \(x^{5}-1\) is divided by \(x-1\) and the quotient when \(x^{6}-1\) is divided by \(x-1 .\) Observe the pattern in the first two quotients and then write the quotient for \(x^{9}-1\) divided by \(x-1\) without dividing.

Simplify each complex fraction. $$\frac{\frac{1}{y+2}-\frac{4}{3 y}}{\frac{3}{y}-\frac{2}{y+3}}$$

In place of each question mark in Exercises \(75-92,\) put an expression that will make the rational expressions equivalent. $$\frac{3}{x-4}=\frac{?}{4-x}$$
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