Graphs of Inequalities (Edexcel IGCSE Further Maths)

Finding Regions using Inequalities

Spec point — 3E

How do I draw inequalities on a graph?

• STEP 1 Draw the line (as if using “=”) for each inequality

  • Use a solid line for ≤ or ≥

    • to indicate the line is included

  • Use a dashed line for < or >

    • to indicate the line is not included

• STEP 2 Decide which side of line is wanted.

  • Below line if "y ≤ ..." or "y < ..."

  • Above line if "y ≥ ..." or "y > ..."

  • To the left of the line if "x ≤ ..." or "x < ..."

  • To the right of the line if "x ≥ ..." or "x > ..."

  • If unsure, use a point that's not on the line as a test

    • Substitute its x and y value into the inequality and check if the inequality is satisfied
    • This will tell you whether or not the inequality holds true on that side of the line

  • It's helpful to indicate the 'correct side' of each line on your sketch

• STEP 3 Choose the region that satisfies all of the inequalities

  • This is the region that is on the correct side of all the lines
  • The exam question will often ask you to shade and/or label the region

Exam Hint

• 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

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

$3x+2y\geq12$ $y<2x$ $x<3$

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:

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

The line $3x+2y\geq12$ 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 $3x+2y\geq12$ (or $y\geq-\frac{3}{2}x+6$), the wanted region is above the line We can check this with the point (0, 0)

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

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

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

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

Interpreting Graphical Inequalities

Spec point — 3E

How do I determine the inequalities if given a region on a graph?

• STEP 1 Write down the equation of each line on the graph

• STEP 2 Remember that lines are drawn with:

  • A solid line for ≤ or ≥ (to indicate line included in region)
  • A dashed line for < or > (to indicate line not included)

• STEP 3 Replace = sign with:

  • ≤ or < if shading below line

  • ≥ or > if shading above line

    • Use a point to test if not sure

How do I find optimal solutions from a region on an inequalities graph?

• A question may ask you to optimise a function for points in the region

  • For example, "For all points in the region with coordinates $( x , y )$, $P=3x-2y$. Find the greatest value of $P$."

• In these questions the inequalities will usually all be ≤ or ≥

  • This means all the points on the lines are included in the region

• The optimal solution (minimum or maximum) will always occur at a point where two lines intersect

  • Substitute the coordinates of the intersection points into the function

  • Choose the point which gives the maximum or minimum value, as required

  • If the question asks for integer solutions

    • Check that the optimal intersection point coordinates are both integers
    • If they are not, the solution will occur at the point with integer coordinates nearest the optimal intersection

Worked Example (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 $\leq$ or $\geq$

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

Write down these inequalities for the final answer

$y\leq-x+7,y\leq6x$* and $y\geq x$

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

(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): $P = 20 ( 0 ) - 3 ( 0 ) = 0$

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

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

Final answer: The greatest value is $P=59.5$