Question 105·Hard·Algebra

Pick one point from any table and substitute its $x$ and $y$ values into $2x - 3y \le 6$ and then into $x + y > 4$. Decide if each resulting statement is true or false.

If any one point in a table makes either inequality false, you can eliminate that whole answer choice. You are looking for a table where all three points pass both tests.

In Desmos, enter the two inequalities exactly as they are: 2x - 3y <= 6 and on the next line x + y > 4. You should see two shaded regions; the solution set of the system is the overlapping (darker) shaded region.

For a given answer choice, create a table in Desmos with columns x1 and y1, and enter the three $(x,y)$ pairs from that table as the rows. Desmos will plot these three points on the same coordinate plane as the inequalities.

Check, for each answer choice, whether all three plotted points lie inside the overlapping shaded region of the two inequalities. The correct answer is the choice whose three points are all in that intersection region and none are outside it.

The system of inequalities is

A point $(x,y)$ is a solution to this system only if both inequalities are true when you plug in $x$ and $y$. For the correct table, all three points listed must satisfy both inequalities.

Take a sample point, such as $(2,3)$, and plug it into each inequality:

• First inequality $2x - 3y \le 6$:
  • $2(2) - 3(3) = 4 - 9 = -5$, and $-5 \le 6$ is true.
• Second inequality $x + y > 4$:
  • $2 + 3 = 5$, and $5 > 4$ is true.

So $(2,3)$ is in the solution set. You will use this same plug-in process to test every point in each answer choice.

Now test the points in each of the first three answer choices. You only need one failure in a table to eliminate that choice.

• Choice A points: $(2,3)$, $(6,1)$, $(0,5)$

  • $(6,1)$ in $2x - 3y \le 6$: $2(6) - 3(1) = 12 - 3 = 9$, and $9 \le 6$ is false, so this table is not correct.

• Choice B points: $(3,2)$, $(5,0)$, $(7,3)$

  • $(5,0)$ in $2x - 3y \le 6$: $2(5) - 3(0) = 10$, and $10 \le 6$ is false, so this table is not correct.

• Choice C points: $(2,4)$, $(4,1)$, $(6,0)$

  • $(6,0)$ in $2x - 3y \le 6$: $2(6) - 3(0) = 12$, and $12 \le 6$ is false, so this table is not correct.

All three of these choices are eliminated because each contains at least one point that fails the first inequality.

The only remaining option is choice D, with points $(1,4)$, $(4,3)$, and $(5,2)$. Check each one:

• $(1,4)$:
  • $2(1) - 3(4) = 2 - 12 = -10$, and $-10 \le 6$ is true;
  • $1 + 4 = 5$, and $5 > 4$ is true.
• $(4,3)$:
  • $2(4) - 3(3) = 8 - 9 = -1$, and $-1 \le 6$ is true;
  • $4 + 3 = 7$, and $7 > 4$ is true.
• $(5,2)$:
  • $2(5) - 3(2) = 10 - 6 = 4$, and $4 \le 6$ is true;
  • $5 + 2 = 7$, and $7 > 4$ is true.

All three points in this table satisfy both inequalities, so the correct answer is the table with $(1,4)$, $(4,3)$, and $(5,2)$ (choice D).