Exercises: Graph Linear Inequalities and Systems

When graphing the inequality $y > 2x - 3$, how should the boundary line $y = 2x - 3$ be drawn?

To determine which half-plane to shade for $3x + 2y \leq 12$, a student substitutes $(0, 0)$:
$3(0) + 2(0) = 0 \leq 12$. True.

What does this tell the student?

A graph shows a solid line through $(0, 3)$ and $(2, 0)$ with the region above and to the left shaded.

Which inequality does this graph represent?

For the inequality $y \leq -2x + 4$, which description is correct?

Graph $2x - y > 4$ using the test-point method. State: (a) whether the boundary is solid or dashed, (b) which test point you chose and what happens when you substitute it, and (c) which side to shade.

Graph $y > 3x$ and describe the solution set.

Note: the line $y = 3x$ passes through the origin. Which test point should you use, and why?

Describe how to graph the system $y \geq x - 1$ and $y \leq -x + 3$, and find the solution region.

(a) Describe the boundary lines and which half-plane each inequality shades.
(b) Verify that $(1, 2)$ is in the solution region by checking both inequalities.

Graph the system $x \geq 0$, $y \geq 0$, and $x + y \leq 4$.

(a) Describe each boundary line and what it represents.
(b) What shape does the feasible region form?
(c) List the vertices of the feasible region.