Example of Graphing an Equality

Example 1

Example 2

More than one inequality?

Maximising and Minimising?

To find the Maximum and Minimum value

Example of Linear Programming

### Graphing Lines

Take for example the line 2x - 3y -6=0. Now let us draw this line using our Coordinate Geometry. Remember from our Junior Certificate that we find where the line cuts the X and Y axes. To do this let x=0 and y=0 respectively. We can find and plot 2 points on the line . They are (0,2) and (3,0).

All the points on the above
line are represented by the equation 2x -3y -6=0. A Linear Inequality
is where all the points on one side of the line are represented. Two
examples would be 2x - 3y>0 and 2x -3y -6<0. If we draw the two
inequalities 2x - 3y -6<=0 and 2x - 3y -6>=0 then because of the equals
sign we include all the points on the line itself .

### Graphing the Inequalities 2x - 3y - 6 >= 0 and 2x - 3y - 6<= 0

All the points on the
Linear Inequality 2x -3y -6>=0 are represented by the region indicated
which is otherwise known as the **half plane**.

Likewise all the points on
the Linear Inequality 2x -3y -6<= 0 are represented by the region
indicated which is also known as the **half plane.**

In general to graph the
region represented by a linear inequality do the following:

* Step 1: * Graph the Line (the
easiest points points to use are where the line cuts the Xand Y axes).

*(i) Select a point on one side of the inequality (usually (0,0) ) and put it into the inequality. If it satisfies the inequality then all the points on that side of the line will satisfy the inequality. That side of the line is the required region.*

**Step 2:**(ii) If it does not satisfy the inequality then all the points on the opposite side of the line to the point selected (0,0) will satisfy the inequality. That side of the line is the required region.

### Example 1

Indicate the set of points
which satisfy the inequality

x + 2y >= 6

### Example 2:

Indicate the set of points
which satisfy the inequality 3x + 4y<=12

* Step 1: *Graph the line
using the points (0,3) and (4,0).

*Select (0,0) and we get 0 + 0 <=12 which is true and therefore satisfys the inequality. Hence it is this side of the line which is the required region.*

**Step 2:**

### More than one inequality?

We are often asked to find a region that is common to more than one inequality. To do this we draw each inequality using a
similar method to the examples above. The we draw a diagram and shade the enclosed region.

### Maximising and Minimising?

When we have found this enclosed region bounded by three or more inequalities it is now possible to find the coordinates of the
Maximum and Minimum points. The Maximum and Minimum points of an enclosed region will always occur at one of the vertices of the
region.

### To find the Maximum and Minimum value

* Step 1:*
Draw the lines and indicate the relevant half planes with arrows.

**Shade the enclosed region and find each vertex.**

*Step 2:***Using simultaneous equations find the other remaining vertices of the common region.**

*Step 3:**Take each of the values in turn and substitute into the rule to find the maximum and minimum value.*

**Step 4:**In the following example we will see how Linear Inequalities can be used in a practical way (Hence the name Linear Programming) to find the maximum or minimum value. We will also see how Linear Programming can provide answers to a wide range of problems within certain constraints. Click below to see a completed example.