Sure, let's address each of the problems step-by-step:
### 1. Property Used: [tex]$(-6) + (-5) = (-5) + (-6)$[/tex]
The property demonstrated here is the Commutative Property of Addition. This property states that the order in which numbers are added does not change the sum. In other words, [tex]\( a + b = b + a \)[/tex]. Thus, [tex]\( (-6) + (-5) = (-5) + (-6) \)[/tex].
### 2. Evaluate: [tex]\( 27 + [(-10) + 1] \)[/tex]
Let's break this down step-by-step:
- First, evaluate the expression inside the brackets: [tex]\( (-10) + 1 = -9 \)[/tex].
- Now, add the result to 27: [tex]\( 27 + (-9) = 27 - 9 = 18 \)[/tex].
So, the evaluation of [tex]\( 27 + [(-10) + 1] \)[/tex] is 18.
### 3. Find [tex]\( 0.135 = 15 \)[/tex]
This appears to be a malformed equation because it states [tex]\( 0.135 = 15 \)[/tex], which is incorrect. There is no meaningful solution as [tex]\( 0.135 \)[/tex] does not equal 15. Hence, this is mathematically false.
### 4. Find the Range of the Data Set: [tex]\( 13, 25, 32, 6, 52, 45 \)[/tex]
To find the range of a data set:
- Identify the maximum value: [tex]\( \max(13, 25, 32, 6, 52, 45) = 52 \)[/tex]
- Identify the minimum value: [tex]\( \min(13, 25, 32, 6, 52, 45) = 6 \)[/tex]
- Calculate the range: [tex]\( \text{Range} = \max - \min = 52 - 6 = 46 \)[/tex]
So, the range of the data set is 46.
### 5. If [tex]\( \frac{2}{3} \)[/tex] of a Number is 6, Find the Number
Let the unknown number be [tex]\( x \)[/tex]:
- According to the problem: [tex]\( \frac{2}{3} x = 6 \)[/tex]
- Solving for [tex]\( x \)[/tex], we multiply both sides by the reciprocal of [tex]\( \frac{2}{3} \)[/tex]:
[tex]\[
x = 6 \cdot \frac{3}{2} = 6 \cdot 1.5 = 9
\][/tex]
Thus, the number is 9.0.
### 6. Between Which Two Numbers Does [tex]\( \frac{11}{4} \)[/tex] Lie?
Let's convert [tex]\( \frac{11}{4} \)[/tex] to a mixed number:
- Dividing 11 by 4 gives us 2 with a remainder of 3, so:
[tex]\[
\frac{11}{4} = 2 \frac{3}{4}
\][/tex]
- Therefore, [tex]\( 2.75 \)[/tex] lies between 2 and 3.
Thus, [tex]\( \frac{11}{4} \)[/tex] lies between the numbers 2 and 3.
