Inequality (science)

Mathematical inequalities play a powerful and widespread role every day and every where. Their utility spans many fields: budgeting in finance, cost and resource analysis in manufacturing, and disaster risk evaluation in actuarial science, all require the diligent application of inequalities.

Inequalities in One Variable

Consider some simple situations in which inequalities occur. Assume that Tom needs to catch a bus at 7:00 in the morning. In order to catch the bus, Tom needs to arrive at the bus stop at or before 6:30 a.m. The inequality represents this requirement, where the variable T is Tom’s arrival time at the bus stop. When Tom buys a bus ticket ($2.20), the amount of money in Tom’s pocket needs to be at least $2.20. This inequality is , where the variable M is the amount of money in Tom’s pocket. As the bus approaches Tom’s destination, Tom may need to alert the driver before the bus reaches his expected stop. If the distance from the preceding stop to Tom’s stop is 1.6 miles, the inequality represents the distance D that Tom travels from the preceding stop before alerting the bus driver. The three inequalities for one variable above share a property: the variable is bounded above, below, or on both sides by some constants. The inequalities give a range restriction to a variable.

Inequalities on a Number Line

Inequalities can be conveniently plotted on the real line. The three inequalities introduced in the previous section can be visually expressed on a number line as follows.

The bracket on the left-hand side of the time point 6:30 a.m. indicates that any arrival time on or before 6:30 a.m. will enable Tom to catch the bus.

The bracket on the right-hand side of the bus fare $2.20 shows that any amount of pocket money more than $2.20 will enable Tom to buy the bus ticket.

The bracket between the preceding bus stop (Hardy Street) and Tom’s destination (Maple Street) indicates the travel distances from Hardy Street where Tom may signal the bus driver to stop at Maple Street.

Compound Inequalities

In the previous three examples, the former two are relatively straightforward; however, the last example is a combination of two simple inequalities associated with two conditions. Tom needs to signal the bus driver after the bus stop at Hardy Street (if he signals too early, he may confuse the bus driver). In this case, Tom’s travel distance from the Hardy Street stop is: . Moreover, Tom needs to make the signal before the Maple Street stop (or the bus driver may pass Maple Street without stopping). In this case, his travel distance from the Hardy Street stop before making the stop signal must be no more than 1.6 miles: miles. Combining the two conditions gives a compound inequality: miles.

Compound inequalities place upper and lower bounds on the variable. What follows is another example of a compound inequality. Assume that Mary has $30, that each pen costs $3.00, and that each notebook costs $2.50. In order to save $10 for lunch, Mary needs to budget the numbers of pens and notebooks that she may buy in the following compound inequality in two variables, , where x and y are the numbers of pens and notebooks, respectively, that Mary may buy. This use of compound inequalities occurs in cost estimation (manufacturing), budgeting (finance), and portfolio allocation analysis (business and investments).

Graphing and Solving Systems of Inequalities

When two or more inequalities are considered, it is essential to know whether all the conditions can be satisfied simultaneously. For instance, it does not make sense to ask Tony to take a train leaving at 6:30 a.m. from Central Station and to buy coffee from a coffeeshop in the same station that opens at 6:40 a.m., because the system of inequalities

where T is the time requirement on Tony, has no solution. In other words, Tony is unable to satisfy these two conditions at the same time. Similarly, checking the plausibility of resource constraints for a certain product necessitates solving a system of inequalities. Graphing is a common solution method.

For instance, consider the following system of inequalities:

Notice that any value from −2 to 2 satisfies the condition of . The three inequalities in the system can be graphed as follows.

From the graph above, the common part among the sections corresponding to the three inequalities in the system is , which means that any values from 1.5 to 2 will satisfy the three conditions in the system.

Graphing and Solving Linear Inequalities

Some situations require working with a system in which the inequality is a linear function of a variable. The approach to solving this type of inequality system is to first simplify each linear inequality and then graph each condition on a number line to find the common section that satisfies all the inequalities. For instance, consider the system

The above system is equivalent to

,

which then simplifies to

The solution to the system can then be graphed as follows.

Thus, any value from 1/3 to 1/2 is a solution to the system of linear inequalities.

Graphing Linear Inequalities in Two Variables

The methods discussed above focus on inequalities in one variable. In this section, we discuss a method to graphing and solving linear inequalities in two variables. Consider the following example:

Due to the involvement of two variables x and y, the graph is in the xy plane, and the shaded area is the solution to the system of linear inequalities in two variables.

Solving Equations and Inequalities Through Substitution

When the system involves equations and inequalities, the solution of the system can be achieved by the graphing method, with a different interpretation. Consider the following system:

The graphing method is as follows.

Notice that by substituting the first equation into the second inequality, the second inequality becomes . Thus the solutions of the system are the points (x, y) satisfying and , as indicated in the red section in the diagram. By the graphing method, the red section is actually the intersection between the equation and the area covered by the two-variable inequality

Probability Inequalities

Aspects of inequalities discussed in the previous sections are non-random inequalities that depend on one or several variables. However, there are other inequalities that have widespread applications in practice, which are probability inequalities. Based on randomness involved, inequalities can be classified into two types—probability inequalities and non-random inequalities. Probability inequalities emerge in applied scenarios, such as "the chance it rains tomorrow is at least 70%" in weather forecasts; "the risk of getting lung cancer for smokers is higher than the risk of non-smokers" in clinical trials; or "the mortality rate of vascular patients is decreasing" in medical investigations. This type of inequality is governed by random factors such as precipitation in weather forecasts, stochastic effects of a stock market in investment portfolio analysis, or the efficacy/toxicity of drug responses in pharmaceutical research.

Depending on the number of events and the level of complexity of a practical situation, probability inequalities can be grouped as univariate inequalities and multivariate inequalities. Well-known univariate probability inequalities include Boole inequality, Bonferoni inequality, Markov inequality, and Chebyshev inequality, among others used in simultaneous statistical inference.

Bibliography

Beckenbach, E., and Bellman, R. An Introduction to Inequalities. Washington, DC: Mathematical Assoc. of America, 1961.

Chen, J. T. Multivariate Bonferroni-Type Inequalities, Theory and Applications. Boca Raton, FL: Chapman, 2014.

Cvetkovski, Zdravko. Inequalities: Theorems, Techniques and Selected Problems. New York: Springer, 2012.