To solve the equation 4x² - 12x = 0, we can first factor out the common factor of 4x:

4x(x - 3) = 0

Now, set each factor equal to zero to solve for x:

4x = 0 => x = 0
x - 3 = 0 => x = 3

Thus, the solution to the equation 4x² - 12x = 0 is x = 0 and x = 3, which corresponds to option C.

To find the inverse of a function, we swap the x and y coordinates (x and f(x), respectively) and then solve for the new f(x):

1. Replace f(x) with y:
y = 3x - 2

2. Swap x and y:
x = 3y - 2

3. Solve for y:
x + 2 = 3y
y = (x + 2) / 3

The inverse of the function f(x) = 3x - 2 is f^(-1)(x) = (x + 2) / 3, which corresponds to option C.

To solve the absolute value equation |3x - 4| = 8, we need to split the equation into two separate equations:

3x - 4 = 8 and 3x - 4 = -8

For the first equation, add 4 to both sides:
3x = 12
Divide by 3:
x = 4

For the second equation, add 4 to both sides:
3x = -4
Divide by 3:
x = -4/3

Thus, the solution to the absolute value equation |3x - 4| = 8 is x = 4 and x = -4/3, which corresponds to option D.

To solve the inequality 2(x - 3) + 4 < 3x + 1, follow these steps: Step 1: Distribute the 2 inside the parentheses: 2x - 6 + 4 < 3x + 1 Step 2: Simplify the inequality: 2x - 2 < 3x + 1 Step 3: Subtract 2x from both sides of the inequality: -2 < x + 1 Step 4: Subtract 1 from both sides of the inequality: -3 < x Thus, the solution to the inequality 2(x - 3) + 4 < 3x + 1 is x > -3, which corresponds to option A.

To solve for y, we will isolate it from the equation, 2x + 3y = 12.

Step 1: Subtract 2x from both sides of the equation
3y = -2x + 12

Step 2: Divide by 3 to isolate y
y = -(2/3)x + 4

So, the equivalent expression is y = -(2/3)x + 4, which corresponds to option A.