To solve the equation [tex]\(\frac{2x}{3} - \frac{1}{3} = \frac{x}{2} + \frac{3}{2}\)[/tex], follow these steps:
1. Clear the Fractions:
The first step is to eliminate the fractions by finding a common multiple of the denominators (3 and 2). The least common multiple (LCM) of 3 and 2 is 6. Therefore, we will multiply every term of the equation by 6 to clear the fractions.
[tex]\[
6 \left(\frac{2x}{3} - \frac{1}{3}\right) = 6 \left(\frac{x}{2} + \frac{3}{2}\right)
\][/tex]
Distribute the 6:
[tex]\[
6 \cdot \frac{2x}{3} - 6 \cdot \frac{1}{3} = 6 \cdot \frac{x}{2} + 6 \cdot \frac{3}{2}
\][/tex]
Simplify each term:
[tex]\[
2 \cdot 2x - 2 = 3x + 3 \cdot 3
\][/tex]
[tex]\[
4x - 2 = 3x + 9
\][/tex]
2. Isolate the Variable:
Now we need to get all the terms involving [tex]\(x\)[/tex] on one side of the equation and the constant terms on the other side. Subtract [tex]\(3x\)[/tex] from both sides:
[tex]\[
4x - 3x - 2 = 9
\][/tex]
Simplify:
[tex]\[
x - 2 = 9
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Finally, add 2 to both sides to isolate [tex]\(x\)[/tex]:
[tex]\[
x - 2 + 2 = 9 + 2
\][/tex]
[tex]\[
x = 11
\][/tex]
Therefore, the solution to the equation [tex]\(\frac{2x}{3} - \frac{1}{3} = \frac{x}{2} + \frac{3}{2}\)[/tex] is:
[tex]\[
x = 11
\][/tex]
