1. **State the problem:** Solve the equation $$\frac{4x - 1}{2} = x + 7$$ for $x$. 2. **Formula and rules:** To solve linear equations, we aim to isolate $x$ on one side by performing inverse operations such as addition, subtraction, multiplication, and division. 3. **Step 1: Eliminate the denominator** by multiplying both sides by 2: $$2 \times \frac{4x - 1}{2} = 2 \times (x + 7)$$ which simplifies to $$4x - 1 = 2x + 14$$ 4. **Step 2: Get all $x$ terms on one side and constants on the other:** Subtract $2x$ from both sides: $$4x - 2x - 1 = 14$$ which simplifies to $$2x - 1 = 14$$ 5. **Step 3: Isolate $x$:** Add 1 to both sides: $$2x = 15$$ 6. **Step 4: Solve for $x$ by dividing both sides by 2:** $$x = \frac{15}{2}$$ 7. **Final answer:** $$x = 7.5$$ This means the solution to the equation is $x = 7.5$.