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Math Lesson 10.4.2 - The Minimum or Maximum Values of a System of Linear Inequalities

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Welcome to our Math lesson on The Minimum or Maximum Values of a System of Linear Inequalities, this is the second lesson of our suite of math lessons covering the topic of Systems of Inequalities, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Determining the Minimum or Maximum Value(s) of a System of Linear Inequalities

Sometimes, we are asked to find the minimum or maximum values of a system of linear inequalities. This means finding one of the following values:

  1. The maximum x-value of the system. In other words, we are asked to find the rightmost value of the solution set on the graph;
  2. The minimum x-value of the system. In other words, we are asked to find the leftmost value of the solution set on the graph;
  3. The maximum y-value of the system. In other words, we are asked to find the uppermost value of the solution set on the graph;
  4. The minimum y-value of the system. In other words, we are asked to find the lowermost value of the solution set on the graph;
  5. A combination of two minimum or maximum coordinates, where possible.

Obviously, the method used for this purpose consists of determining the solution zone by solving the given system of linear inequalities first, and eventually, solving analytically the corresponding system of linear equations to determine the intercept point of the graph with high precision - an action that is often impossible to do through the graphing method, as the solution pair may consist of rational numbers. For example, in the system of linear inequalities

2x + 3 < y3x - y ≥ 4

we arrange first the two inequalities of the system to make them appear in the form y (?) mx + n, where (?) represents one of the four inequality symbols. Thus, for the first inequality, we have

2x + 3 < y

Looking at this inequality from right to left yields

y > 2x + 3

Hence, the solution set of this inequality alone includes all points above the graph without those on the graph line.

On the other hand, for the second inequality, we have

3x - y ≥ 4
3x - y - 3x ≥ 4 - 3x
-y ≥ -3x + 4
(-1) ∙ (-y) ≥ (-1)(-3x + 4)
y ≤ 3x - 4

Therefore, the solution set of this inequality alone includes all points under the graph including those on the graph line as well.

The figure below shows this system of linear inequalities solved graphically.

Math Tutorials: Systems of Inequalities Example

From the above graph, it is evident that the solution set has a minimum y-value, which is at the lowest point of the solution zone. Hence, we can find the minimum y-coordinate of the solution zone by looking at the intercept point of the two graphs. Since the graph method is not very helpful in finding a precise y-coordinate of the intersection, we solve analytically the system of the corresponding linear equations to determine this point with high precision. Thus, we have

2x + 3 = y3x - y = 4

or

2x - y = -33x - y = 4

Multiplying the first equation by (-1) to eliminate y and then adding the two equations yields

(-1) ∙ (2x - y) = (-1) ∙ (-3)3x - y = 4
-2x + y = 33x - y = 4
-2x + 3x + y - y = 3 + 4
x = 7

Now, substituting this value found for x in any of the equations (for example in the second) we are able to find the y-coordinate of the intercept, which represents the minimum value of the solution set for the original system of inequalities. Thus,

3x - y = 4
3 ∙ 7 - y = 4
21 - y = 4
y = 17

Therefore, the minimum value of the system of the inequalities

2x + 3 < y3x - y ≥ 4

is ymin = 17 without including this point, as one of the inequalities (the first) does not include it. We say, "The solution zone includes only values that have their y-coordinate more than 17".

In addition, we can find the leftmost value of the solution set by considering the x-value of the solution set. Thus, x = 7 represents the minimum x-value of the solution set, where again this value is not included, as the first inequality does not contain it.

Example 2

Calculate the maximum or minimum x- and y-values of the solution set in the system of linear inequalities

4x - y ≤ -13x + 5 > 6y

Solution 2

First, we have to express each inequality in the form y (?) mx + n to see whether we have to check for minimum or maximum values in the solution set. Thus, for the first inequality, we have

4x - y ≤ -1
-y ≤ -4x - 1
(-1) ∙ (-y) ≤ (-1) ∙ (-4x - 1)
y ≥ 4x + 1

and for the second inequality, we have

3x + 5 > 6y
6y < 3x + 5
6y/6 < 3x/6 + 5/6
y < x/2 + 5/6

Since the solution zone of the first inequality includes the zone above the graph (including the line) while that of the second inequality includes the zone below the graph (without the line), it is impossible to determine whether the solution zone contains a maximum or minimum without plotting the graphs. Hence, we refer to the figure below for more detailed info.

Math Tutorials: Systems of Inequalities Example

From the figure, it is clear that the solution set has a maximum point at the intercept of the two graph lines made of the uppermost y-coordinate and the rightmost x-coordinate. The graphing method is not very helpful in determining these two coordinates, so we have to solve the system of the corresponding linear equations analytically. Thus, we have

4x - y = -13x + 5 = 6y
4x - y = -13x - 6y = -5

Let's express the first equation as y = 4x + 1 and solve the second equation in terms of x by substituting y with 4x + 1. Thus,

3x - 6(4x + 1) = -5
3x - 24x - 6 = -5
-21x = -5 + 6
-21x = 1
x = -1/21

As you see, this is a rational value, impossible to find directly in the graph. Hence, we say "xmax = -1/21".

As for the maximum y-value of the solution set (ymax), we substitute in the first equation the value found for x, which yields

4x - y = -1
y = 4x + 1
y = 4 ∙ (-1/21) + 1
= -4/21 + 21/21
= 17/21

Thus, the maximum value of the solution zone is ymax = 17/21, as shown in the graph.

More Systems of Inequalities Lessons and Learning Resources

Inequalities Learning Material
Tutorial IDMath Tutorial TitleTutorialVideo
Tutorial
Revision
Notes
Revision
Questions
10.4Systems of Inequalities
Lesson IDMath Lesson TitleLessonVideo
Lesson
10.4.1Systems of Linear Inequalities
10.4.2The Minimum or Maximum Values of a System of Linear Inequalities
10.4.3Systems of Three Linear Inequalities
10.4.4Systems of Inequalities where one inequality is Quadratic and the other is Linear

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