Pick one table and test its pairs one by one: first compute $3x - y$ and compare it to $5$, then compute $x + y$ and compare it to $10$.

Pay special attention to the difference between $\ge$ and $<$: one allows equality, the other does not. How does that affect pairs where $x + y$ equals $10$?

If even one pair in a table fails either inequality, you can cross out that entire table and move on to the next.

For each pair in a table, type an expression like 3*(x_value) - (y_value) (for example, 3*2 - 2 for $(2,2)$). Compare the numerical result Desmos gives you to $5$ and decide whether it is $\ge 5$. If any pair gives a result less than $5$, eliminate that table.

For the same pairs, type (x_value) + (y_value) (for example, 2 + 2 for $(2,2)$). Compare the result to $10$ and check whether it is strictly less than $10$. If any pair gives $10$ or more, eliminate that table.

After checking all three pairs in each table with these two expressions, the correct table is the one where every pair produces a value $\ge 5$ for $3x - y$ and a value $< 10$ for $x + y$ when evaluated in Desmos.

A pair $(x, y)$ is a solution to the system if both inequalities are true when you substitute $x$ and $y$:

• Check $3x - y \ge 5$.
• Check $x + y < 10$.

If a pair fails either inequality, it is not a solution. A table is correct only if all three of its pairs work for both inequalities.

Start by checking $3x - y \ge 5$ for the pairs in each table.

Example with a pair from one table:

• For $(2, 2)$: $3(2) - 2 = 6 - 2 = 4$, and $4$ is not $\ge 5$, so $(2, 2)$ does not satisfy the first inequality. Any table containing $(2, 2)$ cannot be correct.

Do a similar quick check for at least one pair in each table; if a single pair fails $3x - y \ge 5$, you can discard that whole table.

For remaining candidate tables, check the second inequality $x + y < 10$.

Be careful: $<$ means strictly less than, so $x + y = 10$ is not allowed.

• For example, if you find a pair where $x + y = 10$, then that pair fails the second inequality, and its entire table cannot be correct.

After this step, only tables whose pairs all pass both checks should remain.

Take the table that has not been eliminated and verify each of its three pairs carefully:

• For $(2,1)$: $3(2) - 1 = 5 \ge 5$ and $2 + 1 = 3 < 10$.
• For $(3,2)$: $3(3) - 2 = 7 \ge 5$ and $3 + 2 = 5 < 10$.
• For $(4,5)$: $3(4) - 5 = 7 \ge 5$ and $4 + 5 = 9 < 10$.

All three pairs satisfy both inequalities, so the correct table is the one listing $(2,1)$, $(3,2)$, and $(4,5)$.