Systems of Inequalities - Revision Notes

In these revision notes for Systems of Inequalities, we cover the following key points:

• What are systems of linear inequalities?
• How do we identify the solution set (zone) of a system of linear inequalities?
• Which method is the most suitable for solving systems of inequalities?
• How do we prove the correctness of the solution when dealing with systems of inequalities?
• How do you find the minimum/maximum value of the solution set in systems of inequalities?
• How do you find the leftmost/rightmost value of the solution set in systems of inequalities?
• How do we solve systems of three linear inequalities?
• How do we solve systems of inequalities where at least one inequality is not linear?

Systems of Inequalities Revision Notes

Systems of inequalities contain two or more inequalities to solve simultaneously. The first two (analytical) methods used for solving systems of linear equations (elimination and substitution method) cannot be used in systems of linear inequalities, as here we have to find a region of the coordinates plane XOY where both inequalities are true. This region can extend at different orientations, so it is impossible to identify all values that belong to the solution set without referring to a figure that shows the graphs of every single inequality.

When solving only graphically such systems, we must isolate the variable y and solve each inequality in terms of the other variable x/. Then, we plot the graphs of each inequality and see where the double shaded area that shows the region where both inequalities are true does extend.

The graphing method not only makes us solve systems of linear inequalities, but it also helps us solve easier the corresponding systems of equations when the coefficients are such that they give rational numbers as solutions. Thus, instead of finding a pair of values obtained through complicated calculations, we simply plot the graphs and see what the coordinates of the intercept point are.

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 finding such minimum or maximum values 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.

Sometimes, a third inequality is added to the normal systems of two linear inequalities, producing a system of three linear inequalities. The simplest case is when the third inequality contains a single variable and acts as a third boundary line for the solution zone. In this way, we often obtain a closed figure (triangle) as a solution zone, formed by the two-by-two intercepts of the three lines.

However, it is not always possible to obtain a closed region as a solution zone for a system of three linear inequalities. For example, if two of the linear inequalities are dependent (i.e. when they have parallel graphs), we obtain a zone that is limited in two or three directions but unlimited in the fourth, as the graphs form a figure where two parallel lines are intercepted by a third one.

When solving systems of inequalities where not all are linear (for example, when one inequality is quadratic), we use the same approach as in systems of linear inequalities. The only difference is that one of the lines is a parabola, so the solution zone has one curved side.

Another thing to point out in such systems is that if we are interested to find any minimum or maximum point, we must solve the corresponding system of equations only by the substitution method. Thus, we express one of the variables in the linear equation in terms of the other variable, and then we substitute it in the quadratic equation.