Final answer:
To solve the equation 1/2x + 2 (3/4x -1) = 1/4x + 6, we need to simplify both sides of the equation and solve for x. The solution is x = 32/9.
Step-by-step explanation:
To solve the equation 1/2x + 2 (3/4x -1) = 1/4x + 6, we need to simplify both sides of the equation and solve for x. First, distribute the 2 to the terms inside the parentheses: 1/2x + 6/4x - 2 = 1/4x + 6. Next, combine like terms on both sides: 1/2x + 3/2x - 2 = 1/4x + 6. Combining the x terms, we get 5/2x - 2 = 1/4x + 6. To get rid of the fractions, multiply both sides of the equation by 4 to clear the fractions: 4 * (5/2x - 2) = 4 * (1/4x + 6). This gives us 10x - 8 = x + 24. Subtract x from both sides to isolate the x term: 10x - x - 8 = 24. Simplifying this, we have 9x - 8 = 24. Add 8 to both sides: 9x = 32. Finally, divide both sides by 9 to solve for x: x = 32/9.