Question 295·Medium·Algebra

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Consider the system of inequalities

\begin{cases} 3x-2y \ge 6\\[2pt] x+y<5 \end{cases}

For which of the following tables are all the ordered pairs $(x,y)$ listed solutions to the system?

$x$	$y$
4	2
3	0
2	2

$x$	$y$
1	1
2	0
0	3

$x$	$y$
6	2
4	-3
5	1

$x$	$y$
4	0
5	-1
3	1

A system of inequalities asks for ordered pairs that satisfy both conditions. With several tables to test, graph the inequalities in Desmos and plot a pair from each table. One failed pair rules out a table; then check every pair in the table that remains. Watch the word all: finding one pair that works isn't enough.

Hints

Hint 1
In an ordered pair $(x,y)$ , the first number is $x$ and the second is $y$ . A solution must make both inequalities true. What does Desmos show when you graph the two inequalities together?
Hint 2
You can reject a table as soon as one pair fails either inequality. Plot a pair from each table first, then spend time checking every pair only in a table that remains.
Hint 3
The symbol $<$ makes a strict inequality: equality doesn't count. Desmos draws its boundary as a dashed line. The boundary for $\ge$ is solid because equality does count there.

Step-by-step

Graph the overlap and test the tables
Step 1
Find the region that satisfies both conditions
Type $3x-2y\ge6$ and $x+y<5$ into Desmos, each on its own line. Desmos shades the points that satisfy each inequality. Their overlap contains the solutions to both. A boundary line is where equality holds: the line for $\ge$ is solid and included, while the line for $<$ is dashed and excluded.
Step 2
Reject two tables with one pair each
Plot $(4,2)$ from the first table and $(6,2)$ from the third. Desmos puts both outside the shading for $x+y<5$ : their sums are $6$ and $8$ , respectively. One failed pair rules out a whole table. You don't need to test the other pairs in either table.
Step 3
Test a pair from the second table
Plot $(1,1)$ from the second table. Desmos puts it outside the shading for $3x-2y\ge6$ : substituting its coordinates gives $3(1)-2(1)=1$ , which is less than $6$ . It passes $x+y<5$ , but passing only one condition isn't enough.
Step 4
Check every pair in the remaining table
Plot $(4,0)$ , $(5,-1)$ , and $(3,1)$ from the last table. Desmos shows all three in the overlap, not on the excluded dashed line. So every ordered pair in that table satisfies both inequalities. Choice D.

Question 295·Medium·Algebra

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