Question 190·Medium·Algebra

For each answer choice, substitute the $x$- and $y$-values into $2x + 3y < 6$. Which options make the left side less than 6, and which do not?

Pay close attention to the difference between $<$ and $\ge$. A result equal to 6 does not satisfy $2x + 3y < 6$, but a result equal to $x - 4$ does satisfy $y \ge x - 4$.

Once you know which ordered pairs (if any) work for $2x + 3y < 6$, test those in $y \ge x - 4$ by computing $x - 4$ and comparing it with $y$.

In Desmos, enter the inequalities as two separate expressions:

• 2x + 3y < 6
• y >= x - 4

Desmos will shade the region for each inequality; the solution set to the system is where the shaded regions overlap.

Add each answer choice as a point in Desmos by typing them on separate lines, for example:

• (0, 2)
• (2, -1)
• (1, 3)
• (4, 0)

Each will appear as a dot on the graph.

Look for which of the four plotted points lies inside the overlapping shaded region (not just on a region that only one inequality shades). The point that lies in this common region corresponds to the correct answer.

A solution to the system

is an $(x,y)$ pair that makes both inequalities true at the same time. We can test each answer choice by substituting its $x$- and $y$-values into each inequality.

Check $2x + 3y < 6$ for each option:

• For $(0, 2)$: $2(0) + 3(2) = 0 + 6 = 6$, and $6 < 6$ is false, so this pair fails the first inequality.
• For $(2, -1)$: $2(2) + 3(-1) = 4 - 3 = 1$, and $1 < 6$ is true, so this pair passes the first inequality.
• For $(1, 3)$: $2(1) + 3(3) = 2 + 9 = 11$, and $11 < 6$ is false, so this pair fails the first inequality.
• For $(4, 0)$: $2(4) + 3(0) = 8 + 0 = 8$, and $8 < 6$ is false, so this pair fails the first inequality.

Only one of the four choices makes $2x + 3y < 6$ true; now we must see if that same choice also satisfies the second inequality $y \ge x - 4$.

Take the only pair that satisfied $2x + 3y < 6$ and test it in $y \ge x - 4$:

• For $(2, -1)$: here $x = 2$ and $y = -1$.
  • Compute $x - 4$: $2 - 4 = -2$.
  • Compare $y$ and $x - 4$: $-1 \ge -2$ is true.

So $(2, -1)$ satisfies both $2x + 3y < 6$ and $y \ge x - 4$. Therefore, the solution to the system among the answer choices is $(2, -1)$.