Which four inequalities define this region?

Which four inequalities define this region?

Question

$O$ is the origin and $A$, $B$ and $C$ are the points $(0,-1)$, $(2,-1)$ and $(1,0)$ respectively. Write down the four inequalities which define the shaded region $ABCO$ including its boundary.

Suggestion

Can you find the equations of the four lines that bound the shaded area? How can you use these four equations to answer the original question?

Solution

The shaded region is bounded by four lines: $y=0$, $x=0$, $y=-1$ and by the straight line through $(0,1)$, $(1,0)$ and $(2,-1)$.

Writing this straight line as $y=mx+c$, we know that $c=1$, since $c$ is the $y$-intercept. We know the point $(1,0)$ is on the line, so $0=m+1$, which means the fourth line segment bounding the shaded area is part of $y=1-x$.

Since the shaded area is below the $x$-axis, the first inequality is $y\le 0,$ since we wish to include the boundary of the shaded area.

Since the shaded area is to the right of the $y$-axis, the second inequality is $x\ge 0.$ The shaded region is above the line $y=-1$, so the third equation is $y\ge-1.$

Which side of the line $y = 1-x$ do we want? We want the side that includes the origin. When $x = 0$ and $y = 0, y < 1-x$, so this is the side we want.

So the fourth and final inequality is $y\le 1-x.$