Systems of Linear Inequalities in Two Variables

Question 1

The region that satisfies ALL inequalities in a system is called the:

Accepted Answer

Feasible region — The overlap of all shaded regions is the feasible region.

Question 2

A point is a solution of a system of inequalities only if it:

Accepted Answer

Satisfies every inequality in the system — All constraints must hold simultaneously.

Question 3

Does $(2,2)$ satisfy the system $y \leq x+1$ and $y \geq 0$?

Accepted Answer

Yes — $2\leq 2+1=3$ (true) and $2\geq0$ (true), so both hold: yes.

Question 4

Does $(4,1)$ satisfy the system $x + y \leq 4$ and $x \geq 0$?

Accepted Answer

No — $4+1=5\leq4$ is false, so $(4,1)$ is not a solution even though $x\geq0$ holds.

Question 5

To be in the feasible region of a system with $3$ inequalities, a point must satisfy:

Accepted Answer

All 3 — Every inequality must be satisfied at once.

Question 6

The feasible region of the system $x \geq 0$, $y \geq 0$ lies in which quadrant?

Accepted Answer

First quadrant — $x\geq0$ and $y\geq0$ together describe the first quadrant.

Question 7

Which point lies in the feasible region of $y \geq 1$ and $y \leq 3$?

Accepted Answer

$(0,2)$ — Need $1\leq y\leq3$. Only $(0,2)$ has $y=2$ in this range.

Question 8

For the system $y > x$ and $y < 4$, does the point $(1,3)$ belong to the feasible region?

Accepted Answer

Yes — $3>1$ (true) and $3

Question 9

If a system's inequalities have no common region, the feasible region is:

Accepted Answer

Empty (no solution) — With no overlap, no point satisfies all inequalities, so the region is empty.

Question 10

A shop makes $x$ chairs and $y$ tables. "At most $10$ items in total" is written as:

Accepted Answer

$x + y \leq 10$ — "At most $10$ in total" means the sum $x+y$ is not more than $10$: $x+y\leq10$.