Coaching Centre
Regions on the Cartesian plane
Extension: graphing linear inequalities as half-planes, dashed and solid boundaries, testing a point, and regions defined by several inequalities.
Half-planes
Drag P above and below the dashed line y = 2x + 1.
Sketch $y > 2x + 1$.
For $y > 2x + 1$, the boundary line is:
Strict inequality.
Is $(1, 4)$ in $y > 2x + 1$?
$4 > 3$.
Is $(0, 0)$ in $y > 2x + 1$?
$0 > 1$ is false.
Is $(0, 0)$ in $x + y \le 3$?
$0 \le 3$.
$y \ge -x + 3$ is shaded:
$\ge$: solid, and $y$ is greater, so above.
Which point is in both $y > 0$ and $x + y < 4$?
$2 > 0$ and $3 < 4$.
The region $x < 2$ is to the:
$x < 2$: left of $x = 2$, dashed.
Is $(2, 5)$ on the boundary of $y \le 2x + 1$, and is it included?
$2(2) + 1 = 5$; $\le$ includes the line.
New numbers every time. Keep going until you get five in a row.