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Chapter 1: Problem 41

Find (a) the opposite (or additive inverse) of each number and (b) the absolute value of each number. $$ -\frac{3}{4} $$

Short Answer

Expert verified
Opposite: \frac{3}{4}\, Absolute Value: \frac{3}{4}\

Step by step solution

01

Identify the Number

The given number is \(-\frac{3}{4} \).
02

Find the Opposite (Additive Inverse)

The opposite (or additive inverse) of a number is the number that, when added to the original number, results in zero. For \(-\frac{3}{4}\), the opposite is \ \frac{3}{4} \.
03

Find the Absolute Value

The absolute value of a number is the distance from the number to zero on the number line, regardless of direction. For \(-\frac{3}{4}\), the absolute value is \ \frac{3}{4} \.

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Key Concepts

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

Additive Inverse
In algebra, the additive inverse of a number is crucial. The additive inverse is essentially the number that, when added to the original number, returns a sum of zero.
  • For example, the additive inverse of \(-4\) is \(+4\).

  • In our exercise, we have the number \(-\frac{3}{4}\).
To find its additive inverse, we need to find the number that, when added to \(-\frac{3}{4}\), equals zero.
This means that the additive inverse of \(-\frac{3}{4}\) is \(+\frac{3}{4}\). To generalize, if the original number is positive, the additive inverse is the same number with a negative sign, and vice versa.
Absolute Value
The term absolute value refers to how far a number is from zero on the number line, without considering direction.
  • It is always a non-negative number.
  • For instance, the absolute value of \(-5\) is \(+5\), and the absolute value of \(+5\) is also \(+5\).
  • In mathematical notation, absolute value is represented with two vertical bars, such as \(|-5| = 5\).

  • In our exercise, the absolute value of \(-\frac{3}{4}\) is \(|-\frac{3}{4}|= \frac{3}{4}\).
No matter what the original number is, positive or negative, its absolute value is always a positive distance to zero. This concept is very helpful in various algebraic calculations and equations.
Number Line
A number line is a visual representation of numbers laid out in a straight line, typically with zero at the center.
  • Positive numbers lie to the right of zero.

  • Negative numbers lie to the left of zero.

Using a number line helps to understand where a number exists in relation to zero and other numbers.
  • For instance, \(-\frac{3}{4}\) lies to the left of zero, indicating that it is a negative number.

  • On the same number line, \(+\frac{3}{4}\) will be situated to the right of zero.
    • This makes it easier to comprehend concepts like additive inverse and absolute value, as you can easily see how far a number is from zero and what its opposite would be.

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