Question 145·Medium·Algebra

When you substitute a negative $y$-value, remember that subtracting a negative number is the same as adding a positive number.

An ordered pair is only acceptable if it makes both inequalities true; if it fails even one, you can eliminate that choice.

In $2x - y \ge 4$, equality is allowed, but in $x + 3y < 5$, the expression must be strictly less than $5$, not equal to $5$ or greater.

In Desmos, enter 2x - y >= 4 on one line and x + 3y < 5 on another. Desmos will shade the solution region for each inequality; the overlapping shaded area represents points that satisfy both.

For each option, type it as a point, for example (3,1), (0,-2), (1,2), and (2,-1). Desmos will show each point on the same coordinate plane as the shaded regions.

Look to see which of the plotted points lies inside the overlapping (common) shaded region of the two inequalities. That point corresponds to the answer choice that satisfies both inequalities.

You are given two inequalities:

An ordered pair $(x,y)$ satisfies both inequalities if, when you substitute $x$ and $y$ into each inequality, both statements are true. Your job is to see which choice makes both inequalities true at the same time.

Take a generic ordered pair, say $(x,y) = (a,b)$.

• In the first inequality, substitute $x=a$ and $y=b$ to get $2a - b$ and compare it to $4$.
• In the second inequality, substitute $x=a$ and $y=b$ to get $a + 3b$ and compare it to $5$.

The pair $(a,b)$ works only if:

• $2a - b$ is greater than or equal to $4$, and
• $a + 3b$ is less than $5$.

We will do this same process for each answer choice.

Now plug in each of the first three choices.

Choice A: $(3,1)$

• First inequality: $2x - y = 2(3) - 1 = 6 - 1 = 5$, and $5 \ge 4$ is true.
• Second inequality: $x + 3y = 3 + 3(1) = 3 + 3 = 6$, and $6 < 5$ is false. So $(3,1)$ fails the second inequality.

Choice B: $(0,-2)$

• First inequality: $2x - y = 2(0) - (-2) = 0 + 2 = 2$, and $2 \ge 4$ is false. So $(0,-2)$ already fails the first inequality.

Choice C: $(1,2)$

• First inequality: $2x - y = 2(1) - 2 = 2 - 2 = 0$, and $0 \ge 4$ is false. So $(1,2)$ fails the first inequality.

So far, none of these three satisfy both inequalities.

Now test the remaining option.

Choice D: $(2,-1)$

• First inequality: $2x - y = 2(2) - (-1) = 4 + 1 = 5$, and $5 \ge 4$ is true.
• Second inequality: $x + 3y = 2 + 3(-1) = 2 - 3 = -1$, and $-1 < 5$ is true.

This ordered pair makes both inequalities true, so the correct answer is $(2,-1)$.