Compute each multiplication carefully: what is $\tfrac{1}{3} \cdot 9$? What is $\tfrac{1}{3} \cdot (-6x)$? Then do the same for the $\tfrac{1}{2}$ terms.

After distributing, group together the $x$-terms and the constant numbers. Add the coefficients of $x$ and then add the constants to get a single simplified expression.

In Desmos, type y = (1/3)(9 - 6x) + (1/2)(x - 4) - 5 so you can see the graph of the original expression as a line.

On new lines, type y = -3/2 x + 6, y = 3/2 x - 4, y = -1/2 x - 4, and y = -3/2 x - 4. Each will appear as a different line on the graph.

Look for the answer-choice line that lies exactly on top of the original expression’s line for all $x$-values. The choice whose graph completely overlaps the original line is the equivalent expression.

Start by using the distributive property on each set of parentheses:

• For $\tfrac{1}{3}(9 - 6x)$:

  • $\tfrac{1}{3} \cdot 9 = 3$
  • $\tfrac{1}{3} \cdot (-6x) = -2x$ So this part becomes $3 - 2x$.

• For $\tfrac{1}{2}(x - 4)$:

  • $\tfrac{1}{2} \cdot x = \tfrac{1}{2}x$
  • $\tfrac{1}{2} \cdot (-4) = -2$ So this part becomes $\tfrac{1}{2}x - 2$.

Now the whole expression is:

$3 - 2x + \tfrac{1}{2}x - 2 - 5$.

Group the $x$-terms and the constant terms:

• $x$-terms: $-2x + \tfrac{1}{2}x$
• Constants: $3 - 2 - 5$

First, combine the $x$-terms:

$-2x + \tfrac{1}{2}x = -\tfrac{4}{2}x + \tfrac{1}{2}x = -\tfrac{3}{2}x$.

Then, combine the constants:

$3 - 2 - 5 = 1 - 5 = -4$.

So the simplified expression is $-\tfrac{3}{2}x - 4$, which matches choice D.