Let's break down and solve the expression step by step:

\[
\frac{2}{3} \times \left(-\frac{5}{7}\right) + \frac{7}{3} + \frac{2}{3} \times \left(-\frac{2}{7}\right)
\]

### Step 1: Multiply the fractions.
First, we'll perform the multiplications:

\[
\frac{2}{3} \times \left(-\frac{5}{7}\right) = -\frac{10}{21}
\]
\[
\frac{2}{3} \times \left(-\frac{2}{7}\right) = -\frac{4}{21}
\]

### Step 2: Rewrite the expression.
Now, rewrite the expression with the results from the multiplications:

\[
-\frac{10}{21} + \frac{7}{3} - \frac{4}{21}
\]

### Step 3: Find a common denominator.
The common denominator between 21 and 3 is 21. We'll rewrite \(\frac{7}{3}\) with a denominator of 21:

\[
\frac{7}{3} = \frac{49}{21}
\]

Now the expression becomes:

\[
-\frac{10}{21} + \frac{49}{21} - \frac{4}{21}
\]

### Step 4: Combine the terms.
Since the denominators are the same, we can combine the numerators:

\[
\frac{-10 + 49 - 4}{21} = \frac{35}{21}
\]

### Step 5: Simplify the fraction.
\(\frac{35}{21}\) simplifies to:

\[
\frac{5}{3}
\]

### Final Answer:
The simplified value of the expression is:

\[
\frac{5}{3}
\]

Would you like further details or explanations on any part of the solution?

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Here are 5 related questions to further your understanding:
1. How do you find a common denominator between two fractions?
2. Can you explain how to simplify a fraction?
3. What are the rules for multiplying fractions?
4. How do you add and subtract fractions with different denominators?
5. Can you provide another example of simplifying a complex fraction?

**Tip:** When working with fractions, always look for opportunities to simplify as early as possible to make calculations easier.

Discover how to solve the expression 2/3 × -5/7 + 7/3 + 2/3 × -2/7 with detailed steps. This problem involves multiplying and adding fractions, finding common denominators, and simplifying the result. Perfect for students in Grades 6-8.

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