We are given the equation:

\[
7\left( -\frac{1}{2}x - 1 \right) + \frac{4}{5}x = -3x - 1
\]

### Step 1: Distribute the 7 on the left-hand side
Distribute the 7 inside the parentheses:

\[
7 \left( -\frac{1}{2}x \right) + 7(-1) + \frac{4}{5}x = -3x - 1
\]

This simplifies to:

\[
-\frac{7}{2}x - 7 + \frac{4}{5}x = -3x - 1
\]

### Step 2: Combine like terms
Now let's combine the terms with \(x\) on the left-hand side and isolate the constant terms:

\[
-\frac{7}{2}x + \frac{4}{5}x = -3x - 1 + 7
\]

Simplify the right-hand side:

\[
-\frac{7}{2}x + \frac{4}{5}x = -3x + 6
\]

### Step 3: Eliminate fractions by finding a common denominator
To eliminate the fractions, the least common denominator (LCD) of 2 and 5 is 10. Multiply every term by 10:

\[
10 \left( -\frac{7}{2}x \right) + 10 \left( \frac{4}{5}x \right) = 10(-3x) + 10(6)
\]

This gives us:

\[
-35x + 8x = -30x + 60
\]

### Step 4: Simplify the equation
Now, combine the like terms:

\[
-27x = -30x + 60
\]

### Step 5: Isolate \(x\)
Add \(30x\) to both sides to isolate \(x\):

\[
3x = 60
\]

Finally, divide both sides by 3:

\[
x = 20
\]

### Solution:
The solution to the equation is:

\[
x = 20
\]

Would you like further details or explanations?

Here are 5 related questions for further exploration:
1. How do you solve equations with variables on both sides?
2. What is the best method to handle fractions in algebraic equations?
3. How can we check if the solution to a linear equation is correct?
4. What are common mistakes to avoid when solving multi-step equations?
5. How do you solve linear equations involving decimals?

**Tip:** When working with fractions in equations, clearing the fractions by multiplying through by the least common denominator (LCD) can simplify the process.

Learn how to solve the linear equation 7(-1/2x - 1) + 4/5x = -3x - 1 with fractions. This detailed step-by-step solution covers important algebraic concepts such as the distributive property, combining like terms, and eliminating fractions using the least common denominator (LCD). Ideal for students in Grades 8-10.

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