To simplify the expression [tex]\(\left(\frac{2a + 4b}{3}\right) - \left(\frac{3(a-b)}{2}\right)\)[/tex], follow these detailed steps:
1. Distribute and Simplify Within Each Fraction
Left Fraction:
[tex]\[
\frac{2a + 4b}{3}
\][/tex]
There's no further simplification needed within this fraction.
Right Fraction:
[tex]\[
\frac{3(a - b)}{2} = \frac{3a - 3b}{2}
\][/tex]
2. Find a Common Denominator
The fractions have denominators of 3 and 2, respectively. The least common multiple (LCM) of 3 and 2 is 6. Therefore, we need to express each fraction with a denominator of 6.
Left Fraction:
[tex]\[
\frac{2a + 4b}{3} \times \frac{2}{2} = \frac{2(2a + 4b)}{6} = \frac{4a + 8b}{6}
\][/tex]
Right Fraction:
[tex]\[
\frac{3(a - b)}{2} \times \frac{3}{3} = \frac{3(3a - 3b)}{6} = \frac{9a - 9b}{6}
\][/tex]
3. Subtract the Two Fractions
Now that both fractions have a common denominator, we can subtract them directly.
[tex]\[
\frac{4a + 8b}{6} - \frac{9a - 9b}{6} = \frac{(4a + 8b) - (9a - 9b)}{6}
\][/tex]
4. Combine Like Terms in the Numerator
Distribute and combine like terms inside the numerator:
[tex]\[
4a + 8b - 9a + 9b = (4a - 9a) + (8b + 9b) = -5a + 17b
\][/tex]
5. Simplify the Overall Expression
Place the simplified numerator over the common denominator:
[tex]\[
\frac{-5a + 17b}{6}
\][/tex]
Thus, the simplified form of the expression [tex]\(\left(\frac{2a + 4b}{3}\right) - \left(\frac{3(a - b)}{2}\right)\)[/tex] is:
[tex]\[
\boxed{\frac{-5a + 17b}{6}}
\][/tex]
