To solve the equation:

\[
0.15 = 110 + \frac{\frac{1000 - x}{6}}{\frac{1000 + x}{3}}
\]

we'll follow these steps:

### Step 1: Simplify the complex fraction

The right-hand side contains a fraction within a fraction. To simplify this, we'll first handle the fraction inside the larger fraction.

\[
\frac{\frac{1000 - x}{6}}{\frac{1000 + x}{3}} = \frac{1000 - x}{6} \times \frac{3}{1000 + x}
\]

Simplify this by multiplying:

\[
= \frac{3(1000 - x)}{6(1000 + x)} = \frac{1000 - x}{2(1000 + x)}
\]

Now, the equation becomes:

\[
0.15 = 110 + \frac{1000 - x}{2(1000 + x)}
\]

### Step 2: Isolate the fraction term

Subtract 110 from both sides:

\[
0.15 - 110 = \frac{1000 - x}{2(1000 + x)}
\]

Simplify the left-hand side:

\[
-109.85 = \frac{1000 - x}{2(1000 + x)}
\]

### Step 3: Eliminate the denominator

Multiply both sides by \( 2(1000 + x) \) to get rid of the fraction:

\[
-109.85 \cdot 2(1000 + x) = 1000 - x
\]

Simplify:

\[
-219.7(1000 + x) = 1000 - x
\]

### Step 4: Expand both sides

Expand the left-hand side:

\[
-219700 - 219.7x = 1000 - x
\]

### Step 5: Combine like terms

Move all terms involving \( x \) to one side and constants to the other:

\[
-219.7x + x = 1000 + 219700
\]

\[
-218.7x = 220700
\]

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

Divide both sides by -218.7:

\[
x = \frac{220700}{-218.7} \approx -1009.14
\]

### Final Answer:
\[
x \approx -1009.14
\]

Let me know if you need further clarification or more details on any step.

---

Here are 5 questions you might explore next:

1. What happens if we change the constants in the equation?
2. Can you simplify complex fractions like this in other situations?
3. How would you solve if the fraction involved different variables?
4. Could we solve this equation graphically to understand the behavior?
5. How does multiplying by denominators affect equation solving?

**Tip:** Always isolate variables by simplifying fractions and eliminating denominators early in the process to make solving easier.

0.15 = 110 + [(1000 - x) / 6] / [(1000 + x) / 3]

Explore a detailed solution to the equation 0.15 = 110 + [(1000 - x) / 6] / [(1000 + x) / 3], involving rational expressions and complex fractions. This solution includes simplifying nested fractions, isolating variables, and solving linear equations. Ideal for students in Grades 9-12 or anyone learning advanced algebra.

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