We are solving the equation:

$\frac{2x+1}{4} - x = \frac{3x+1}{8} + 1$

Step 1: Eliminate the fractions

Find the least common denominator (LCD) of $4$ and $8$, which is $8$. Multiply through the entire equation by $8$ to eliminate the fractions:

$8 \cdot \frac{2x+1}{4} - 8 \cdot x = 8 \cdot \frac{3x+1}{8} + 8 \cdot 1$

Simplify each term:

$2(2x + 1) - 8x = 3x + 1 + 8$

$4x + 2 - 8x = 3x + 9$

Step 2: Simplify the equation

Combine like terms:

$-4x + 2 = 3x + 9$

Step 3: Isolate $x$

Move $3x$ to the left-hand side and $2$ to the right-hand side:

$-4x - 3x = 9 - 2$

$-7x = 7$

Step 4: Solve for $x$

Divide both sides by $-7$:

$x = -1$

Final Answer:

$x = -1$

Would you like me to verify the solution or explain any specific step in more detail?

Here are some related questions for practice:

1. How do you solve linear equations with fractions?
2. What is the purpose of finding the least common denominator?
3. How can we check if a solution is correct by substitution?
4. What happens if a fraction has variables in both numerator and denominator?
5. How does multiplying through by the LCD simplify solving equations?

Tip: When solving equations with fractions, always simplify by eliminating fractions early to make calculations easier!