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.