Answer :
- Sort the constants: $13 < 15$.
- List the factors of $13(x+7)$: 1, 13, $(x+7)$, $13(x+7)$.
- List the factors of $15(x+5)$: 1, 3, 5, 15, $(x+5)$, $3(x+5)$, $5(x+5)$, $15(x+5)$.
- Sort the expressions based on complexity: $13, 15, x+7, 13(x+7)$.
### Explanation
1. Understanding the Problem
We are given the expressions $x+7$, 13, 15, $13(x+7)$, factors of $13(x+7)$, and factors of $15(x+5)$. The goal is to sort these expressions. However, the instructions are ambiguous. Sorting numbers is straightforward, but sorting expressions involving a variable $x$ requires additional context or a specific criterion. Also, 'factors of an expression' are not directly comparable to the expressions themselves. Therefore, I will make some assumptions to provide a meaningful sort. I will assume that we want to sort the expressions $x+7$, 13, 15, and $13(x+7)$ based on their potential values for a given $x$. I will also list the factors of $13(x+7)$ and $15(x+5)$.
2. Analyzing the Expressions
Let's analyze the given expressions:
1. $x+7$: This is a linear expression. Its value depends on the value of $x$.
2. $13$: This is a constant. Its value is always 13.
3. $15$: This is a constant. Its value is always 15.
4. $13(x+7)$: This is also a linear expression. Its value depends on the value of $x$.
We can sort the constants easily: $13 < 15$.
To compare the linear expressions with the constants, we need to consider different values of $x$. However, without a specific range for $x$, we can only make general observations.
3. Listing the Factors
Let's list the factors of $13(x+7)$ and $15(x+5)$:
Factors of $13(x+7)$: 1, 13, $(x+7)$, $13(x+7)$.
Factors of $15(x+5)$: 1, 3, 5, 15, $(x+5)$, $3(x+5)$, $5(x+5)$, $15(x+5)$.
4. Sorting the Expressions
Now, let's sort the expressions $x+7$, 13, 15, and $13(x+7)$ based on their complexity. We can consider constants as simpler than linear expressions. Among the constants, we can sort them numerically. Among the linear expressions, we can sort them based on their leading coefficients. However, this is just one possible way to sort them.
One possible sorting is: 13, 15, $x+7$, $13(x+7)$.
5. Final Answer
Final Answer: The sorted expressions are 13, 15, $x+7$, $13(x+7)$. The factors of $13(x+7)$ are 1, 13, $(x+7)$, $13(x+7)$. The factors of $15(x+5)$ are 1, 3, 5, 15, $(x+5)$, $3(x+5)$, $5(x+5)$, $15(x+5)$.
### Examples
Sorting expressions is a fundamental concept in algebra and is used in various applications. For example, when simplifying algebraic expressions or solving equations, it's essential to organize terms and factors in a logical order. In computer science, sorting algorithms are used to arrange data in a specific order, which can improve the efficiency of searching and processing information. Understanding how to sort expressions helps in optimizing processes and making calculations more manageable.
- List the factors of $13(x+7)$: 1, 13, $(x+7)$, $13(x+7)$.
- List the factors of $15(x+5)$: 1, 3, 5, 15, $(x+5)$, $3(x+5)$, $5(x+5)$, $15(x+5)$.
- Sort the expressions based on complexity: $13, 15, x+7, 13(x+7)$.
### Explanation
1. Understanding the Problem
We are given the expressions $x+7$, 13, 15, $13(x+7)$, factors of $13(x+7)$, and factors of $15(x+5)$. The goal is to sort these expressions. However, the instructions are ambiguous. Sorting numbers is straightforward, but sorting expressions involving a variable $x$ requires additional context or a specific criterion. Also, 'factors of an expression' are not directly comparable to the expressions themselves. Therefore, I will make some assumptions to provide a meaningful sort. I will assume that we want to sort the expressions $x+7$, 13, 15, and $13(x+7)$ based on their potential values for a given $x$. I will also list the factors of $13(x+7)$ and $15(x+5)$.
2. Analyzing the Expressions
Let's analyze the given expressions:
1. $x+7$: This is a linear expression. Its value depends on the value of $x$.
2. $13$: This is a constant. Its value is always 13.
3. $15$: This is a constant. Its value is always 15.
4. $13(x+7)$: This is also a linear expression. Its value depends on the value of $x$.
We can sort the constants easily: $13 < 15$.
To compare the linear expressions with the constants, we need to consider different values of $x$. However, without a specific range for $x$, we can only make general observations.
3. Listing the Factors
Let's list the factors of $13(x+7)$ and $15(x+5)$:
Factors of $13(x+7)$: 1, 13, $(x+7)$, $13(x+7)$.
Factors of $15(x+5)$: 1, 3, 5, 15, $(x+5)$, $3(x+5)$, $5(x+5)$, $15(x+5)$.
4. Sorting the Expressions
Now, let's sort the expressions $x+7$, 13, 15, and $13(x+7)$ based on their complexity. We can consider constants as simpler than linear expressions. Among the constants, we can sort them numerically. Among the linear expressions, we can sort them based on their leading coefficients. However, this is just one possible way to sort them.
One possible sorting is: 13, 15, $x+7$, $13(x+7)$.
5. Final Answer
Final Answer: The sorted expressions are 13, 15, $x+7$, $13(x+7)$. The factors of $13(x+7)$ are 1, 13, $(x+7)$, $13(x+7)$. The factors of $15(x+5)$ are 1, 3, 5, 15, $(x+5)$, $3(x+5)$, $5(x+5)$, $15(x+5)$.
### Examples
Sorting expressions is a fundamental concept in algebra and is used in various applications. For example, when simplifying algebraic expressions or solving equations, it's essential to organize terms and factors in a logical order. In computer science, sorting algorithms are used to arrange data in a specific order, which can improve the efficiency of searching and processing information. Understanding how to sort expressions helps in optimizing processes and making calculations more manageable.
