Think about what operation will isolate $2x + y$ on the left side. What should you do to both sides of the inequality?

When you divide or multiply an inequality by a negative number, what must you do to the inequality sign? Keep that in mind when simplifying.

Once you have an inequality in terms of $2x + y$ alone, match its sign ($\le$ or $\ge$) and constant term with the answer choices.

In Desmos, enter the inequality -4(2x + y) <= 8. You will see a shaded region representing all $(x,y)$ pairs that satisfy the original inequality.

On separate lines, enter each choice (for example, 2x + y <= -2, -2x - y >= 2, etc.). For each one, compare its shaded region to the region from the original inequality.

The correct choice is the one whose shaded region exactly overlaps the region from -4(2x + y) <= 8—no extra points and no missing points. Read off which inequality that is from your Desmos list.

Two inequalities are equivalent if they have exactly the same solution set. That means any $(x,y)$ pair that makes the original inequality true must also make the new one true, and vice versa.

So our goal is to simplify the given inequality without changing its solutions.

Start with the given inequality:

First, you can distribute $-4$ or go directly to isolating the expression in parentheses. Let's isolate $2x + y$ by undoing the multiplication by $-4$.

The expression $2x + y$ is being multiplied by $-4$, so divide both sides by $-4$:

When you divide both sides of an inequality by a negative number, you must reverse (flip) the inequality sign from $\le$ to $\ge$.

Now simplify each side:

• Left side: $\frac{-4(2x + y)}{-4} = 2x + y$
• Right side: $\frac{8}{-4} = -2$

After dividing and simplifying, you get:

Now compare this to the answer choices and select the one that matches exactly: $2x + y \ge -2$, which is choice D.