Dealing With More Than One Inequality
If more than one inequality has to be satisfied, then the required region will have more than one boundary. The diagram below shows the inequalities x≥1, y≥1 and x+y≤6.
The triangle indicated by bold lines has been shaded three times. The points inside this region, including those points on each of the boundaries, satisfy all three inequalities.
📌 Worked Example 1
Find the region which satisfies the inequalities x≤4, y≤2x, y≥x+1.
Write down the coordinates of the Vertices of this region.
First shade the region which is satisfied by the inequality x≤4.
Then add the region which satisfies
using a different type of shading, as shown.
Finally, add the region which is satisfied by
using a third type of shading.
The region which has been shaded in all three different ways (the triangle outlined in bold) satisfies all three inequalities.
The coordinates of its Vertices can be seen from the diagram as (1, 2), (4, 5) and (4,8).
When a large number of inequalities are involved, and therefore a greater amount of shading, the required region becomes more difficult to see on the graph.
Therefore it is better to shade out rather shade in, leaving the required region unshaded. This method is used in the following example, where ’shadow’ shading indicates the side of the line which does not satisfy the relevant inequality.
📌 Worked Example 2
A small factory employs people at two rates of pay. The maximum number of people who can be employed is 10. More workers are employed on the lower rate than on the higher rate.
Describe this situation using inequalities, and draw a graph to show the region in which they are satisfied.
Let x= number employed at the lower rate of pay, and y= number employed at the higher rate of pay.
The maximum number of people who can be employed is 10, so x+y≤10.
As more people are employed at the lower rate than the higher rate, then x>y.
As neither x nor y can be negative, then x≥0 and y≥0.
These inequalities are represented on the graph below.
The triangle formed by the unshaded sides of each line is the region where all four
inequalities are satisfied. The dots indicate all the possible employment options. Note that only integer values inside the region are possible solutions.
📌 Worked Example 3
Find the linear inequalities for which the shaded region in the given figure is the solution set.
(i) Consider 2x+3y=3. We observe that the shaded region and the origin lie on opposite side of this line and (0, 0) satisfies 2x+3y≤3. Therefore, we must have 2x+3y≥3 as linear inequality corresponding to the line 2x+3y=3.
(ii) Consider 3x+4y=18. We observe that the shaded region and the origin lie on the same side of this line and (0, 0) satisfies 3x+4y≤18. Therefore, 3x+4y≤18 is the linear inequality corresponding to the line 3x+4y=18.
(iii) Consider -7x+4y=14. It is clear from the figure that the shaded region and the origin lie on the same side of this line and (0, 0) satisfies the inequality -7x+4y≤14. Therefore, -7x+4y≤14 is the inequality corresponding to the line -7x+4y=14.
(iv) Consider x-6y=3. It may be noted that the shaded portion and origin lie on the same side of this line and (0, 0) satisfies x-6y≤3. Therefore, x-6y≤3 is the inequality corresponding to the line x-6y=3.
(v) Also the shaded region lies in the first quadrant only. Therefore, x≥0, y≥0.
Hence, in view of (i), (ii), (iii), (iv) and (v) above, the linear inequalities corresponding to the given solution set are :
2x+3y≥3, 3x+4y≤18, -7x+4y≤14, x-6y≤3, x≥0, y≥0.