To simplify the expression \( \frac{1}{2}(a - 2b) + \frac{3}{2}(b - 2a) \), let's follow these steps:

### Step 1: Distribute the fractions
Distribute both \( \frac{1}{2} \) and \( \frac{3}{2} \) inside the parentheses:

\[
\frac{1}{2}(a - 2b) = \frac{1}{2}a - \frac{1}{2}(2b) = \frac{1}{2}a - b
\]

\[
\frac{3}{2}(b - 2a) = \frac{3}{2}b - \frac{3}{2}(2a) = \frac{3}{2}b - 3a
\]

### Step 2: Combine the simplified terms
Now add the two expressions:

\[
\left( \frac{1}{2}a - b \right) + \left( \frac{3}{2}b - 3a \right)
\]

### Step 3: Group like terms
Group the terms with \( a \) and the terms with \( b \):

\[
\left( \frac{1}{2}a - 3a \right) + \left( -b + \frac{3}{2}b \right)
\]

### Step 4: Simplify each group
For the \( a \)-terms:

\[
\frac{1}{2}a - 3a = \frac{1}{2}a - \frac{6}{2}a = \frac{-5}{2}a
\]

For the \( b \)-terms:

\[
-b + \frac{3}{2}b = \frac{-2}{2}b + \frac{3}{2}b = \frac{1}{2}b
\]

### Final Expression
Now combine the results:

\[
\frac{-5}{2}a + \frac{1}{2}b
\]

Thus, the simplified expression is:

\[
\frac{-5}{2}a + \frac{1}{2}b
\]

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Would you like more details or have any questions about the steps? Here are 5 related questions for further practice:

1. Simplify \( \frac{2}{3}(x - y) + \frac{1}{3}(y - x) \).
2. Factor the expression \( \frac{-5}{2}a + \frac{1}{2}b \).
3. Simplify \( 2a - (3a + 5b) \).
4. Expand and simplify \( \frac{1}{4}(2x - 3y) + \frac{1}{2}(4y - x) \).
5. Simplify \( \frac{1}{3}(6x - 9y) + \frac{1}{2}(4y - 3x) \).

**Tip**: When simplifying algebraic expressions, always distribute the coefficients first, then combine like terms step by step to avoid mistakes.

Learn how to simplify the algebraic expression 1/2(a-2b) + 3/2(b-2a) using the distributive property and combining like terms. Follow a detailed step-by-step solution suitable for students in Grades 7-9.

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