Let's solve the equation step by step:

Given:
\[
x - \frac{5}{2} - 2x - \frac{3}{5} = \frac{1}{2}
\]

### Step 1: Combine like terms
Combine the \(x\) terms on the left-hand side:
\[
x - 2x = -x
\]
So the equation becomes:
\[
-x - \frac{5}{2} - \frac{3}{5} = \frac{1}{2}
\]

### Step 2: Combine the constants
We need to add the fractions \(-\frac{5}{2}\) and \(-\frac{3}{5}\). To do this, find a common denominator. The least common denominator (LCD) of 2 and 5 is 10.

Rewrite the fractions:
\[
-\frac{5}{2} = -\frac{25}{10}, \quad -\frac{3}{5} = -\frac{6}{10}
\]
Now combine them:
\[
-\frac{25}{10} - \frac{6}{10} = -\frac{31}{10}
\]
So the equation becomes:
\[
-x - \frac{31}{10} = \frac{1}{2}
\]

### Step 3: Eliminate the fraction by multiplying by the LCD
The least common denominator of 10 and 2 is 10. Multiply the entire equation by 10 to eliminate the fractions:
\[
10(-x) - 10\left(\frac{31}{10}\right) = 10\left(\frac{1}{2}\right)
\]
Simplifying:
\[
-10x - 31 = 5
\]

### Step 4: Solve for \(x\)
Add 31 to both sides:
\[
-10x = 5 + 31
\]
\[
-10x = 36
\]
Now divide both sides by -10:
\[
x = \frac{36}{-10} = -\frac{18}{5}
\]

So, the solution is:
\[
x = -\frac{18}{5}
\]

Would you like further explanation or details?

Here are 5 related questions to expand your understanding:
1. How do you solve equations involving fractions?
2. What is the least common denominator and how is it found?
3. How can you check if the solution is correct?
4. How do you handle negative coefficients in equations?
5. What methods can be used to simplify complex algebraic expressions?

**Tip:** Always try to combine like terms first when solving equations—it simplifies the process significantly.

This page provides a step-by-step solution to the linear equation x - 5/2 - 2x - 3/5 = 1/2, demonstrating how to combine like terms and handle fractions. A great resource for students in Grades 7-9 learning algebraic techniques.

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