To determine the distance between the fraction [tex]\(\frac{3}{4}\)[/tex] and the number 2, we can follow these steps:
1. Identify the Values:
- The first value is the fraction [tex]\(\frac{3}{4}\)[/tex].
- The second value is the number 2.
2. Calculate the Distance:
- The distance between two numbers on a number line is given by the absolute difference between them. This is represented by the absolute value function [tex]\( |\cdot| \)[/tex], which converts any number to its non-negative form.
3. Set Up the Absolute Difference:
- Subtract [tex]\(\frac{3}{4}\)[/tex] from 2: [tex]\( 2 - \frac{3}{4} \)[/tex].
- Alternatively, subtract 2 from [tex]\(\frac{3}{4}\)[/tex]: [tex]\(\frac{3}{4} - 2\)[/tex].
4. Apply the Absolute Value:
- Take the absolute value of each expression to ensure the result is non-negative:
[tex]\[
| 2 - \frac{3}{4} |
\][/tex]
[tex]\[
| \frac{3}{4} - 2 |
\][/tex]
5. Evaluate the Expressions (the distance calculation itself):
- Calculate the operations inside the absolute values:
[tex]\[
2 - \frac{3}{4} = 2 - 0.75 = 1.25
\][/tex]
[tex]\[
\frac{3}{4} - 2 = 0.75 - 2 = -1.25
\][/tex]
- Convert each result to its absolute value:
[tex]\[
| 1.25 | = 1.25
\][/tex]
[tex]\[
| -1.25 | = 1.25
\][/tex]
So, the two expressions that represent the distance between [tex]\(\frac{3}{4}\)[/tex] and 2 are:
[tex]\[
| 2 - \frac{3}{4} |
\][/tex]
[tex]\[
| \frac{3}{4} - 2 |
\][/tex]
Both expressions evaluate to [tex]\(1.25\)[/tex], which correctly represents the distance between [tex]\(\frac{3}{4}\)[/tex] and 2.
