Sure, let's solve the equation step-by-step:
Given:
[tex]\[
\frac{x - 3}{5} - \frac{x + 2}{3} = \frac{x}{2} - \frac{1}{3}
\][/tex]
First, find a common denominator for all the terms. The denominators are 5, 3, and 2. The least common multiple (LCM) of these numbers is 30. Therefore, we will multiply each term by 30 to clear the denominators.
1. Multiply both sides of the equation by 30:
[tex]\[
30 \left(\frac{x - 3}{5}\right) - 30 \left(\frac{x + 2}{3}\right) = 30 \left(\frac{x}{2}\right) - 30 \left(\frac{1}{3}\right)
\][/tex]
2. Distribute 30 to each term inside the parentheses:
[tex]\[
30 \cdot \frac{x - 3}{5} - 30 \cdot \frac{x + 2}{3} = 30 \cdot \frac{x}{2} - 30 \cdot \frac{1}{3}
\][/tex]
3. Simplify each term:
[tex]\[
6(x - 3) - 10(x + 2) = 15x - 10
\][/tex]
4. Distribute the constants:
[tex]\[
6x - 18 - 10x - 20 = 15x - 10
\][/tex]
5. Combine like terms on the left side:
[tex]\[
-4x - 38 = 15x - 10
\][/tex]
6. Add [tex]\(4x\)[/tex] to both sides to start isolating [tex]\(x\)[/tex]:
[tex]\[
-38 = 19x - 10
\][/tex]
7. Add 10 to both sides:
[tex]\[
-28 = 19x
\][/tex]
8. Divide both sides by 19:
[tex]\[
x = \frac{-28}{19}
\][/tex]
Hence, the solution to the equation is:
[tex]\[
x \approx -1.47368421052632
\][/tex]
