Let's solve the equation step by step:

Given:
\[
2x = -1 + 8(2x + 1)
\]

1. **Expand the right-hand side**:
   Distribute the 8 across the terms inside the parentheses:
   \[
   2x = -1 + 8(2x) + 8(1)
   \]
   Simplifying this gives:
   \[
   2x = -1 + 16x + 8
   \]
   
2. **Combine like terms on the right-hand side**:
   \[
   2x = 16x + 7
   \]

3. **Move all terms involving \(x\) to one side**:
   Subtract \(16x\) from both sides to isolate the \(x\) terms on one side:
   \[
   2x - 16x = 7
   \]
   Simplifying this:
   \[
   -14x = 7
   \]

4. **Solve for \(x\)**:
   Divide both sides by \(-14\) to get:
   \[
   x = \frac{7}{-14}
   \]
   Simplifying the fraction:
   \[
   x = -\frac{1}{2}
   \]

So, the solution is \( x = -\frac{1}{2} \).

Would you like more details on any of the steps? Here are five related questions to explore further:

1. How can you verify the solution to a linear equation?
2. What are the properties of equality used in solving linear equations?
3. How do you solve equations involving fractions?
4. What is the general strategy for solving equations with variables on both sides?
5. How do different operations affect the equality of an equation?

**Tip:** When dealing with equations, always check your solution by substituting it back into the original equation. This ensures the solution satisfies the equation.

Learn how to solve the linear equation 2x = -1 + 8(2x + 1) step-by-step with a detailed solution. This problem covers key concepts including linear equations, the distributive property, and solving equations involving variables. Suitable for students in Grades 7-9.

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