Q.Solve the system $y > 2x - 3$ and $y \le -x + 4$ graphically. State which boundary line is solid, which is dashed, and describe the direction of shading for each.

[!TLDR] $y=2x-3$ is dashed (strict, shaded above); $y=-x+4$ is solid (slack, shaded at-or-below); the common region is the wedge between them. [!ANSWER] Dashed line $y=2x-3$, shade above it; solid line $y=-x+ …

For $y>2x-3$: the boundary $y=2x-3$ is dashed since the inequality is strict. Testing the origin: is $0>2(0)-3=-3$? Yes, true — so shade the side containing the origin, which is the region above this line (since for any fixed $x$, a larger $y$ means a point drawn higher, i.e. above the line). For $y\le -x+4$: the boundary $y=-x+4$ is solid since the inequality is slack. Testing the origin: is $0\le -0+4=4$? Yes, true — so shade the side containing the origin, the region at or below this line. The two boundary lines intersect where $2x-3=-x+4$, i.e. $3x=7$, $x=\dfrac{7}{3}$, and then $y=2\left(\dfrac{7}{3}\right)-3=\dfrac{14}{3}-\dfrac{9}{3}=\dfrac{5}{3}$. The common (overl …

Classify each inequality's boundary as solid or dashed, test the origin in each original inequality separately to fix the shading …

A common error is shading "above" or "below" a line purely by its equation's sign of slope rather than by actually substituting the test poi …