  \dfrac{ x  }{ 0.5  }  -( \dfrac{ x  }{ 0.3  }  - \dfrac{  \dfrac{ x  }{ 0.6  }  )  }{  \dfrac{ 5  }{ 3  }    }   =  - \dfrac{ 0.1  }{ 0.02  }   

We will solve the equation step by step.

The equation is:

\[
\frac{x}{0.5} - \left( \frac{x}{0.3} - \frac{\frac{x}{0.6}}{\frac{5}{3}} \right) = -\frac{0.1}{0.02}
\]

### Step 1: Simplify the right-hand side

The right-hand side is:

\[
-\frac{0.1}{0.02} = -5
\]

### Step 2: Simplify the inner fraction on the left-hand side

First, simplify \(\frac{\frac{x}{0.6}}{\frac{5}{3}}\).

\[
\frac{\frac{x}{0.6}}{\frac{5}{3}} = \frac{x}{0.6} \times \frac{3}{5} = \frac{3x}{5 \times 0.6} = \frac{3x}{3} = x
\]

Now, substitute this back into the equation:

\[
\frac{x}{0.5} - \left( \frac{x}{0.3} - x \right) = -5
\]

### Step 3: Simplify the terms

Now simplify the term inside the parentheses:

\[
\frac{x}{0.3} - x = \frac{x}{0.3} - \frac{0.3x}{0.3} = \frac{x - 0.3x}{0.3} = \frac{0.7x}{0.3}
\]

Now the equation becomes:

\[
\frac{x}{0.5} - \frac{0.7x}{0.3} = -5
\]

### Step 4: Simplify the left-hand side

\[
\frac{x}{0.5} = 2x \quad \text{and} \quad \frac{0.7x}{0.3} = \frac{7x}{3}
\]

Thus, the equation becomes:

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

### Step 5: Eliminate fractions by multiplying by 3

Multiply the entire equation by 3:

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

This simplifies to:

\[
6x - 7x = -15
\]

### Step 6: Solve for \(x\)

\[
-x = -15
\]

\[
x = 15
\]

### Final Answer: 

\[
x = 15
\]

Would you like any further explanation or details on this solution?

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Here are 5 related questions:
1. How do you simplify fractions with variables in the numerator and denominator?
2. What is the process for multiplying and dividing fractions?
3. How do you solve linear equations with fractions?
4. How can you check if your solution to an equation is correct?
5. What are common mistakes when simplifying complex fractions?

**Tip**: When working with fractions in equations, multiplying both sides by the least common denominator (LCD) can simplify the process significantly!

\dfrac{x}{0.5} - (\dfrac{x}{0.3} - \dfrac{\dfrac{x}{0.6}}{\dfrac{5}{3}}) = -\dfrac{0.1}{0.02}

Learn how to solve a linear equation with complex fractions, including step-by-step simplification of terms and final solution. This problem involves key algebraic concepts, such as fraction operations and linear equation solving, suitable for students in Grades 9-12.

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