Graphs of Inequalities (Edexcel IGCSE Further Pure Maths): Revision Note

• You can also indicate a region by shading the unwanted bits and leaving the region unshaded

  • Some students find this easier

  • The mark scheme awards full marks for either method

On the axes given below show, by shading on your sketch, the region that satisfies the following three inequalities:

$3x+2y\ge 12$                  $y<2x$                 $x<3$

Label the region $R$.

First draw the three straight lines, $3x+2y=12$, $y=2x$ and $x=3$
You may wish to rearrange $3x+2y=12$ to the form $y=mx+c$ first:

$\begin{array}{rcl}2y& =& -3x+12\\ y& =& -\frac{3}{2}x+6\end{array}$

The line $3x+2y\ge 12$ takes a solid line because of the "≥"
The lines $y<2x$ and $x<3$ take dotted lines because of the "<"

Now we need to determine the wanted and unwanted regions

For $3x+2y\ge 12$ (or $y\ge -\frac{3}{2}x+6$), the wanted region is above the line
We can check this with the point (0, 0)

$3\left(0\right)+2\left(0\right)\ge 12$ is false, so (0, 0) does not lie in the wanted region for $\begin{array}{rcl}3x+2y& \ge & 12\end{array}$

For $y<2x$, the wanted region is below the line
If unsure, check with another point, for example (1, 0)

$0<2\left(1\right)$ is true, so (1, 0) lies in the wanted region for $\begin{array}{rcl}y& <& 2x\end{array}$

For $x<3$, the wanted region is to the left of $x=3$ 
(If unsure, you could check with a point)

Finally, shade the region that satisfies all three inequalities on the graph
Don't forget to label the region R

(a) Write down the three inequalities that define the shaded region in the diagram below.

Start by determining the equations of the three lines on the graph.

The line through points (0, 0) and (1, 6) has equation  $y=6x$
The line through points (0, 0) and (2, 2) has equation  $y=x$
The line through points (0, 7) and (7, 0) has equation  $y=-x+7$

Label the lines on the diagram with these equations

The lines are solid, so all the inequalities will be $\le$ or $\ge$

The region is below the lines  $y=-x+7$  and  $y=6x$, so those will be $\le$
The region is above the line  $y=x$, so that will be $\ge$

Write down these inequalities for the final answer

$\mathit{y}\mathbf{\le }\mathbf{-}\mathit{x}\mathbf{+}\mathbf{7}\mathbf{,}\mathit{y}\mathbf{\le }\mathbf{6}\mathit{x}$  and  $\mathit{y}\mathbf{\ge }\mathit{x}$   

For all points in the shaded region, with coordinates $\left(x,y\right)$,  $P=20x-3y$.

(b) Find the greatest value of $P$.

The greatest (and least) values of $P$ will occur at one of the intersections of the lines bordering the region

(These points can be seen on the graph, but it would be worth substituting the coordinates into the equations to make sure)

So we need to find the value of $P$ at each of those points

For point (0, 0):  $P=20\left(0\right)-3\left(0\right)=0$

For point (1, 6):  $P=20\left(1\right)-3\left(6\right)=2$

For point (3.5, 3.5):  $P=20\left(3.5\right)-3\left(3.5\right)=59.5$

The greatest value is $\mathit{P}\mathbf{=}\mathbf{59}\mathbf{.}\mathbf{5}$