Linear Inequalities and Linear Programming

Learning Outcomes

I can solve a linear inequality

I can graph a linear inequality

I can find the region defined by a number of inequalities

I can use the vertices of this region to solve problems

InequalitiesSolution of Linear

Inequalities

Solve – illustrate the solution on a number line

1. x – 2 > 6

2. 3x – 4 > 5

3. 1 – 3x ≤ 5

4. 1 – 3x ≥ 4x – 2

5. 2 < 2x – 1 < 3x + 7

6. 5x – 1 < 4x < 2(x + 1)

InequalitiesGraphical Representation

of Inequalities

Steps

1. Draw the line required for ≥ or ≤ Solid Line > or < Dashed Line

2. Take a test point on one side of the line and test the inequality

3. Shade unwanted region

0

2

4

6

-6

-4

-2

5 10-10 -5

InequalitiesGraphical Representation

of Inequalities

Show on the graph below the region defined by the inequality

y ≤ x + 2

0

2

4

6

-6

-4

-2

5 10-10 -5

InequalitiesGraphical Representation

of Inequalities

Show on the graph below the region defined by the inequality

y ≥ 2x - 3

0

2

4

6

-6

-4

-2

5 10-10 -5

InequalitiesGraphical Representation

of Inequalities

Show on the graph below the region defined by the inequality

y < 4

0

2

4

6

-6

-4

-2

5 10-10 -5

InequalitiesGraphical Representation

of Inequalities

Show on the graph below the region defined by the inequality

x > -6

InequalitiesGraphical Representation

of Inequalities

Give the inequality that defines the unshaded region

0

2

4

-2

2-2

6

x = 0

y = 0

Inequalities Using the Solution Set

The solution set shows all possible solutions to a set of inequalities. Often we are asked to maximise or minimise a particular function using the solution set. To do so we need only consider the vertices (corners) of the solution set.

Example

The technology department in a school is to install two new types of machines.

Machine A costs £2500 each and requires 12m2 floor space; machine B costs £6000 each and requires 6m2 floor space. The spending for the department totals £30,000 and the floor space available for the new machines is 72m2.

Let x be the number of machine A installed. Let y be the number of machine B installed.

Constraints Machine A Machine B Total

Money

Space

Inequalities Example

(a) Show that x and y satisfy the inequality 5x + 12y ≤ 60

(b) Show that x and y satisfy the inequality 2x + y ≤ 12

(c) There are to be at least two of each machine. Write down a further two inequalities.

Inequalities Example

6

8

10

12

0

-2

4

5 10

(d) Illustrate the four inequalities by a suitable diagram on the graph below.

Identify the region containing the set points satisfying all four inequalities. Write the letter R in this region and identify the four vertices.

Inequalities Example

(e) If the maximum number of machine B are installed, how many of machine A is it possible to install?

(f) i. What is the largest number of machines that can be installed?

ii. If the largest number of machines are installed, what is the least floor area that could be used for machines?

Inequalities Additional Notes

Linear Inequalities and Linear Programming

Learning Outcomes:At the end of the topic I will be able to

Can Revise Do Further

I can solve a linear inequality

I can graph a linear inequality

I can find the region defined by a number of inequalities

I can use the vertices of this region to solve problems