To demonstrate how to work through the most complex questions on Homework 5, consider this standard curriculum archetype:
$$\frac{1}{2}(4x - 6) + 3x = 8x - (x + 9)$$
Step 1: Distribute on both sides.
On the left side, multiply $\frac{1}{2}$ by both $4x$ and $-6$. This yields $2x - 3$. On the right side, distribute the negative sign through the parentheses, turning $(x + 9)$ into $-x - 9$.
The equation is now: $2x - 3 + 3x = 8x - x - 9$.
Step 2: Combine like terms on each side.
On the left side, combine $2x$ and $+3x$ to get $5x$. The expression becomes $5x - 3$. On the right side, combine $8x$ and $-x$ to get $7x$. The expression becomes $7x - 9$.
The equation is now: $5x - 3 = 7x - 9$.
Step 3: Move variables to one side.
Subtract $5x$ from both sides of the equation to keep the variable coefficient positive:
$-3 = 7x - 5x - 9$
$-3 = 2x - 9$.
Step 4: Isolate the constant.
Add $9$ to both sides of the equation using inverse operations:
$-3 + 9 = 2x$
$6 = 2x$.
Step 5: Divide by the coefficient.
Divide both sides by $2$:
$\frac{6}{2} = x$
$x = 3$.
Step 6: Verify the result.
Plug $x = 3$ back into the original statement:
Left side: $\frac{1}{2}(4(3) - 6) + 3(3) = \frac{1}{2}(12 - 6) + 9 = \frac{1}{2}(6) + 9 = 3 + 9 = 12$.
Right side: $8(3) - (3 + 9) = 24 - 12 = 12$.
Both sides balance at $12$. The solution is algebraically certain.
