Review question

# Which four inequalities define this region? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R5737

## Solution

$O$ 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.

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 $y\le 1-x.$