Let's break down the expression step by step:

$- \frac{3}{14} + -\frac{5}{17} + -\frac{11}{14} + \frac{1}{-3} + 2 \frac{5}{17}$

We can simplify it into smaller parts:

1. First term: $-\frac{3}{14}$
2. Second term: $-\frac{5}{17}$
3. Third term: $-\frac{11}{14}$
4. Fourth term: $\frac{1}{-3} = -\frac{1}{3}$
5. Fifth term: $2 \frac{5}{17}$ which is a mixed number, so we convert it into an improper fraction: $2 \frac{5}{17} = \frac{34}{17} + \frac{5}{17} = \frac{39}{17}$

Now, we combine all the terms: $-\frac{3}{14} - \frac{5}{17} - \frac{11}{14} - \frac{1}{3} + \frac{39}{17}$

Step 1: Combine terms with the same denominator

• The terms $-\frac{3}{14}$ and $-\frac{11}{14}$ have a common denominator of 14: $-\frac{3}{14} - \frac{11}{14} = -\frac{14}{14} = -1$

• The terms $-\frac{5}{17}$ and $\frac{39}{17}$ have a common denominator of 17: $-\frac{5}{17} + \frac{39}{17} = \frac{34}{17} = 2$

Step 2: Combine the remaining terms

Now the expression looks like: $-1 - \frac{1}{3} + 2$

We can simplify this: $-1 + 2 = 1$ So the expression becomes: $1 - \frac{1}{3}$

Step 3: Final Simplification

To combine $1$ and $-\frac{1}{3}$, we write $1$ as $\frac{3}{3}$, so: $\frac{3}{3} - \frac{1}{3} = \frac{2}{3}$

Final Answer:

$\frac{2}{3}$

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Would you like more details or have any questions about the steps? Here are some related questions you might find useful:

1. How do you combine fractions with different denominators?
2. What are mixed numbers, and how do you convert them to improper fractions?
3. How do you handle negative fractions in addition and subtraction?
4. What is the procedure for simplifying complex rational expressions?
5. Can you show how to add fractions with a denominator of 17 and 14?

Tip: When working with fractions, always find the least common denominator to combine terms easily!