1. **State the problem:** Solve the equation $4 \cdot 2x - 1 = 10 + 2x - 8$ for $x$. 2. **Rewrite the equation:** The equation is $4 \times 2x - 1 = 10 + 2x - 8$. 3. **Simplify both sides:** Left side: $4 \times 2x - 1 = 8x - 1$ Right side: $10 + 2x - 8 = 2 + 2x$ So the equation becomes: $$8x - 1 = 2 + 2x$$ 4. **Isolate variable terms on one side:** Subtract $2x$ from both sides: $$8x - 1 - 2x = 2 + 2x - 2x$$ $$6x - 1 = 2$$ 5. **Isolate constant terms:** Add $1$ to both sides: $$6x - 1 + 1 = 2 + 1$$ $$6x = 3$$ 6. **Solve for $x$ by dividing both sides by 6:** $$x = \frac{3}{6}$$ 7. **Simplify the fraction:** $$x = \frac{\cancel{3}}{\cancel{6}} = \frac{1}{2}$$ **Final answer:** $$x = \frac{1}{2}$$