Question 16·Easy·Linear Inequalities in One or Two Variables

After simplifying once, you may want to change the signs of all the terms. What happens to the inequality sign when you multiply or divide both sides by $-1$?

Most answer choices start with a positive $2x$ term. Once you have a simplified inequality, how can you rewrite it so that it starts with $2x$ while still remaining equivalent?

In Desmos, type the original inequality: -8x + 12y <= 24. You will see a boundary line and a shaded region representing all $(x,y)$ that satisfy the inequality.

On separate lines, enter each answer choice exactly as written (for example, 2x - 3y >= -6, 2x - 3y <= -6, etc.). For each one, look at the boundary line and the shaded region.

Compare the graphs: the correct answer will have exactly the same boundary line and shaded region as the graph of -8x + 12y <= 24. Any choice whose shading is on the opposite side of the line or whose line is in a different place is not equivalent.

The given inequality is

All three terms $-8x$, $12y$, and $24$ are multiples of $4$ (and also of $-4$). This means we can simplify the inequality by dividing every term by the same nonzero number.

To keep track of the inequality sign more easily, start by dividing both sides by $4$ (a positive number):

which simplifies to

Dividing by a positive number does not change the direction of the inequality sign.

The simplified inequality $-2x + 3y \le 6$ is equivalent to the original one, but most of the answer choices are written with a positive $2x$ term. To turn $-2x + 3y$ into $2x - 3y$, multiply every term in the inequality by $-1$:

This will change the signs of all terms.

Multiplying by $-1$ gives

When you multiply or divide an inequality by a negative number, you must reverse the inequality sign, so $\le$ becomes $\ge$:

This is the inequality that is equivalent to the original one, so the correct choice is $2x - 3y \ge -6$.