9
Solving Mixed Fraction Expression -1/2 - (-7/4) ÷ 3/2 + 11/3 × (-5)

To solve the mixed fraction expression \(-\frac{1}{2} - \left(-\frac{7}{4}\right) \div \frac{3}{2} + \frac{11}{3} \times \left(-5\right)\), we'll follow the order of operations, which is parentheses, exponents, multiplication and division (from left to right), and then addition and subtraction (from left to right).

### Step 1: Handle the Negative Signs and Subtraction
Rewrite the expression to simplify the negative signs:
\[
-\frac{1}{2} + \frac{7}{4} \div \frac{3}{2} + \frac{11}{3} \times (-5)
\]

### Step 2: Division
Next, solve the division \(\frac{7}{4} \div \frac{3}{2}\).

Dividing fractions is equivalent to multiplying by the reciprocal:
\[
\frac{7}{4} \div \frac{3}{2} = \frac{7}{4} \times \frac{2}{3} = \frac{7 \times 2}{4 \times 3} = \frac{14}{12} = \frac{7}{6}
\]
So, the expression becomes:
\[
-\frac{1}{2} + \frac{7}{6} + \frac{11}{3} \times (-5)
\]

### Step 3: Multiplication
Now, solve the multiplication \(\frac{11}{3} \times (-5)\).

Multiply the numerator by \(-5\):
\[
\frac{11}{3} \times (-5) = \frac{11 \times (-5)}{3} = \frac{-55}{3}
\]
The expression now is:
\[
-\frac{1}{2} + \frac{7}{6} - \frac{55}{3}
\]

### Step 4: Find a Common Denominator
To add and subtract these fractions, find a common denominator. The least common denominator (LCD) of \(2\), \(6\), and \(3\) is \(6\).

- Convert \(-\frac{1}{2}\) to \(-\frac{3}{6}\).
- \(\frac{7}{6}\) remains as it is.
- Convert \(-\frac{55}{3}\) to \(-\frac{110}{6}\).

The expression now is:
\[
-\frac{3}{6} + \frac{7}{6} - \frac{110}{6}
\]

### Step 5: Combine the Fractions
Add and subtract the fractions with a common denominator:
\[
\frac{-3 + 7 - 110}{6} = \frac{-106}{6}
\]

### Step 6: Simplify the Fraction
Simplify \(\frac{-106}{6}\) by dividing both the numerator and denominator by 2:
\[
\frac{-106}{6} = \frac{-53}{3}
\]

So, the solution to the expression is:
\[
\boxed{\frac{-53}{3}}
\]

Would you like more details or have any questions about this solution? Here are some related questions:

1. How do you multiply mixed fractions with whole numbers?
2. What are some strategies for finding common denominators?
3. Can you explain the steps to convert a negative mixed fraction to an improper fraction?
4. What is the difference between adding and subtracting fractions with different denominators?
5. How can you simplify complex fractions involving multiple operations?

**Tip:** Always simplify fractions as early as possible to make calculations easier.

Learn how to solve the mixed fraction expression -1/2 - (-7/4) ÷ 3/2 + 11/3 × (-5) with a step-by-step solution. This problem involves fractions, negative numbers, and follows the order of operations (PEMDAS/BODMAS). Suitable for students in Grades 7-9.

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