Social inequality encompasses relatively long-lasting differences between groups of people and has considerable implications for individuals, especially "for the rights or opportunities they exercise and the rewards or privileges they enjoy" (Grabb, 1997, pp. 1–2). Most studies of social inequality consider gaps in income and assets between advantaged and disadvantaged groups of people. However, important considerations in the study of social inequality also include issues of status, power, housing, and health, as well as the relationship between these factors and economic well-being.
Economic well-being, status, power, housing, and health are influenced by many factors, including age. In most research on inequality in later life researchers organize age-based systems of inequality on the basis of whether one is old, middle aged, or young. Comparisons are made between these age groups, or strata, to determine whether one group is disadvantaged relative to another.
Other factors that influence social inequality include gender, class, ethnicity, and race. Increasingly, researchers are recognizing that to fully explore the complexities of inequality, all of these factors must be considered. Yet, there remains disagreement among scholars about how class, age, gender, ethnicity, and race influence one another, and about which among these is most important to inequality. In the literature on aging, researchers have approached this issue in various ways. Central to these approaches are debates between multiple-jeopardy and leveling hypotheses, studies of heterogeneity and cumulative advantage/disadvantage, and discussions of diversity.
Multiple bases of inequality: conceptual issues
Research on the relationship between inequality and class, gender, age, ethnicity, and race has often addressed the question of whether the disadvantages associated with class, gender, ethnicity, and race increase in older age (multiple-jeopardy hypotheses), stay the same, or whether the gap between these groups diminishes in later life (leveling hypotheses). Multiple-jeopardy hypotheses argue that class, racial, ethnic, and gender inequalities carry on in later life and that groups who are disadvantaged by these dimensions of inequality in midlife (e.g., women or people of color) face increasing disadvantage in older age. Alternatively, the leveling hypothesis suggests that people who are privileged in early life have more to lose in later life, and that a leveling out of resources takes place. Furthermore, social security policies serve to enhance the resources of those who are disadvantaged earlier in life, which further reduces the gap between the haves and the have-nots among older people. There has been mixed support for both types of hypotheses, depending on what dimensions of inequality are considered and the particular inequality outcome that is of concern. However, after reviewing many diverse studies that consider either the combination of class and age, gender and age, or race and ethnicity and age, Fred Pampel (1998) concluded that there is more support for the multiple-jeopardy hypothesis than for the leveling approach.
Related to multiple-jeopardy arguments are life-course discussions of heterogeneity and cumulative advantage/disadvantage. The key distinction between these two perspectives is the latter's emphasis on time. According to the cumulative advantage/disadvantage hypothesis, the heterogeneity between groups of people increases over time. This hypothesis suggests that individuals have specific class, gender, and racial/ethnic characteristics that provide them with a certain amount of advantage or disadvantage. Initially, there is little separation between the haves and the have-nots on the basis of these distinctions. However, as time passes, the separation between the advantaged and disadvantaged grows and age cohorts become increasingly heterogeneous. The reason that this occurs is because the economic and social value that is attached to productive work in most Western societies differs depending on one's gender, race/ethnicity, class, and age. For the most part, research on cumulative advantage/disadvantage has found support for the hypothesis, but most of the research has focused on income inequality rather than on status, power, housing, or health.
Recent theoretical work on aging and inequality has also considered the issue of diversity. Although the terms heterogeneity and diversity have been used interchangeably, the distinction between them is an important one. Both of these concepts are about difference, but the theoretical emphasis on power relations varies between the two. While heterogeneity can refer to any meaningful group or individual difference, diversity is about "examining groups in relation to interlocking structural positions within a society" (Calasanti, 1996, p. 148). This requires that class, age, gender, and ethnicity/race be conceptualized as "sets of social relations, characterized by power, that are fundamental structures or organizing features of social life" (McMullin, 2000, p. 525), and not as individual attributes. Furthermore, social class, age, gender, and ethnicity/race must be viewed as interlocking sets of power relations. This emphasis on power relations in diversity research prompts a consideration of what is meant by power, and by the related concept status.
Status and power
Put simply, status refers to one's position in society. Sociologists often make a distinction between ascribed status and achieved status. Ascribed status refers to characteristics of individuals over which they have little control, such as their sex, age, race, or ethnicity. Achieved status, on the other hand, refers to positions in society that an individual may achieve, such as level of education or occupational status. Although ascribed and achieved status have received considerable research attention, the status group concept is more relevant to the discussion presented here, because it is directly tied to issues of power. A status group refers to a set of people who "have a subjective sense of common membership and a group awareness that is relatively well defined" (Grabb, 1997, p. 50). Status groups are clearly demarcated in society on the basis of their prestige, honor, and the resulting power that they have.
What then, is meant by power ? There are many sociological definitions of power but perhaps the most widely cited view reflects the ability of individuals or groups in social relationships to impose their will on others regardless of resistance (Weber, 1922). Economic resources are often thought to be associated with power. However, as the preceding discussion of status groups suggests, power may also be derived from prestige and honor.
Because status moves away from the idea that power is largely based in economic relationships, it is tempting to consider issues of aging and inequality in this light. This is because most older people do not have direct associations with the economy (i.e. they do not work for pay), and therefore must derive their status and power, if they have any, from other means. Hence, one question that arises is whether age groups might well be evaluated with regard to their status, and, in turn, whether honor, prestige, and power vary across age groups.
The extent to which age is associated with power and status varies depending on the specifics of the culture or society in question. In some societies, high levels of status and power are found in older age groups relative to younger groups, but in other societies older adults are afforded very little status or power.
In early twenty-first-century Canada and the United States, the relationship between age, status, and power is complex. In these societies, very rarely are young children or teenagers afforded higher levels of status and power than are adults. Regardless of levels of maturity, dexterity, or intellectual ability, teenagers who live in North America must turn a certain chronological age before they are legally able to drive a car, vote, or consume alcohol. Wage scales are established for teenagers, not on the basis of what they do, but on their chronological age. Hence, a twenty year old and a seventeen year old could be working at the same job and the seventeen year old could legally be paid less for doing the same work. All of this suggests that the status and power of younger people in North America lags far behind that of middle-aged or older persons. This assessment is too simplistic, however. A counter pressure to these facts exists which has been referred to as the "cult of youth." This refers to the ideology in North American culture that favors young over old and suggests that to be young is to be vibrant, beautiful, and happy, whereas to be old is to be tired, unattractive, and grim. These cultural views do little to take away from the status and power that middle-aged adults have. However, for older adults these views are especially detrimental because, combined with the loss of their youthful appeal, they have also lost the power and status associated with middle-aged activities such as working for pay and raising families.
The status and power associated with age is influenced by other dimensions of inequality, including class, gender, and ethnicity/race. Yet researchers have not tended to consider these intersections in their discussions of status and power. Anecdotally, however, we can imagine that an older, retired, white man who had a successful career as the president of a large multinational company will likely maintain some status and power despite his age. One way in which men in these circumstances do this is by retaining their memberships on the executive boards of companies or public organizations (e.g., universities). Alternatively, there would be very little status and power associated with being an aging, Hispanic housewife.
As this discussion on status and power illustrates, research on aging tends to insufficiently deal with the intersections between class, age, gender, and ethnicity/race. Where possible, the ensuing discussions will present information on class, age, gender, and ethnicity/race, while maintaining a focus on aging and later life. Notably, very little research has simultaneously and systematically considered all of these dimensions of inequality, thus limiting the extent to which the intersection of these factors can be discussed. Further, most of the available information on health and housing assesses income differences rather than social-class differences and focuses on black/white racial comparisons to the neglect of other racial and ethnic groups. In the following discussion, therefore, income will be used as a proxy for social class and racial comparisons will be limited to discussions of older black and white Americans.
Contrary to popular opinion, the majority (95 percent in the United States) of older people (age 65 and over) live within the community and not in institutional settings such as nursing homes. Among those older people who live in communities, most live in homes that they own (77 percent in the United States).
The structural factors that the U.S. government uses to evaluate housing problems include whether a household has adequate heating and cooling systems, plumbing, and kitchen facilities; whether it has structural defects; and whether there is equipment in need of repair. These structural issues are a significant concern for older people living within a community, because almost 8 percent of them live in dwellings that have moderate to severe structural problems. The U.S. government also assesses housing on the basis of whether it is overcrowded, and on whether there are excessive shelter costs. Very few older adults live in overcrowded households, but one-quarter of older homeowners and one-third of older renters are considered to spend too much of their income on housing. In other words, between one-quarter and one-third of older people spend over 30 percent of their income on housing, and people in this situation are not thought to have enough money available to spend on food, medical expenses, transportation, and clothing.
Low-income households are at higher risk of housing problems than are high-income households, regardless of age. This fact has led some commentators to conclude that housing policies that target older adults should be abandoned for housing policies that focus more specifically on income. Others argue that older adults are in unique situations that require specific housing-policy initiatives. For instance, to the extent that older adults are more frail than younger adults, their housing deficiencies may not be well captured by typical measures of housing problems. As an example, a significant housing problem for an older adult may be the need for assistive devices installed in the bathroom. If an older adult cannot afford such devices, then they are at risk of injury and their home does not meet their basic safety needs. This is an issue that most younger adults do not face. Furthermore, older adults may suffer the negative consequences of living in substandard housing more so than younger adults do. The fact that every year there are North American news reports of older people dying from the summer heat or the winter cold is a point that attests to this claim.
There are also subgroups of older people, the very old (eighty-five+) and black older adults in particular, that are at heightened risk of occupying poor housing (Golant & LaGreca, 1995).This suggests that researchers must carefully consider how age and race intersect in assessments of housing deficiencies and that diversity among those aged sixty-five and older is an important issue in this regard. Assessments of gender differences are rare in the literature on housing deficiencies in later life. This seems a bit odd in light of the fact that higher proportions of older women than older men live alone and that living alone likely increases one's risk of suffering negative consequences of poor housing because there may be no one available to help in times of crisis. Furthermore, because there are well established relationships between poverty and gender and between poverty and poor housing, neglecting potential gender differences in housing seems particularly odd. However, studies on housing often use the household as the unit of analysis and rely on census information that defines household heads on the basis of the name that appears on the lease or land title. Hence, problems with data may make it difficult to fully capture gender inequality in housing.
According to the World Health Organization, health is broadly defined as "a state of complete physical, mental, and social well-being and not merely the absence of disease or infirmity" (World Health Organization, 2000). This definition suggests that many measures of health must be explored to fully understand health and inequality. Older persons are more likely than younger persons to experience chronic health problems, functional impairments, and death. This being said, it is inaccurate to assume that all, or even most, older people are sick to the point that they have trouble functioning. Although the majority (85 percent) of adults age sixty-five or older have one or more chronic illness, only about 20 percent of older adults have trouble providing their own personal care or living independently as a result of functional impairment. Nonetheless, among the older population, health does vary on the basis of age, with those over the age of eighty-five having significantly poorer health than those between the ages of sixty-five and eighty-five.
In general, research suggests that rates of clinical mental disorder are higher among younger than among older adults (Krause, 1999). On the other hand, rates of cognitive impairment and suicide are higher among older than among younger adults (Krause, 1999). Unlike these findings from the psychiatric perspective, sociological research on mental health tends to focus on depressive symptomatology rather than clinical assessments of mental disorder. This research, although still inconclusive, suggests that the relationship between age and depressive symptomatology is nonlinear, decreasing from young to middle adulthood and then increasing at age 60 and thereafter (Krause, 1999).
Gender significantly influences health in later life. On average, women live seven to eight years longer than men. However, older women experience more chronic illnesses and functional impairments, report more depressive symptoms, experience higher levels of psychological distress, and have higher rates of prescription drug use than do older men. Notably, while there is a higher prevalence of depressive disorders among women at all ages when compared to men, this gap decreases with increasing age.
Beyond these age and gender differences, there is further diversity in physical health among older adults on the basis of class and race. One well-known and consistent finding is the relationship between socioeconomic status (SES) and health. Research on physical health generally shows that individuals from lower socioeconomic strata have worse health than do those from higher socioeconomic strata. In general, however, class differences in health are smaller in older age than they are in younger age. Indeed, when education and income are used as measures of SES, the inverse relationship between SES and health is not always supported among samples of older adults. Yet, if occupation is the SES measure used, class differences in health in later life are usually found. For instance, older persons who were previously employed in skilled, white-collar work have fewer health problems and lower rates of mortality than those who were employed in unskilled, blue-collar work (Pampel, 1998).
Regarding mental health, older persons with lower levels of education and income tend to report more depressive symptoms than do those with higher levels of education and income. In contrast, relationships between either education or income and depressive disorder in later life appear to be weak.
Such a strong and consistent relationship between SES and health exists because people with lower levels of income, lower levels of education, and bad jobs are more likely to experience malnutrition, to disproportionately lack knowledge of health care practices, and because they are more often exposed to dangerous working and living environments. All of these things negatively affect health status. Furthermore, research has shown that SES is associated with access to health care services. In particular, older people with higher incomes and who have private health insurance go to the doctor more often and spend more nights in the hospital than do older people who have lower incomes or who do not have private health insurance, regardless of their overall health (Mutchler & Burr, 1991).
The relationship between race and health in later life is complex. For the most part, research suggests that, compared to older white Americans, older black Americans report more chronic health problems, have higher levels of functional decline, and have higher rates of mortality. Yet, even though physical and mental health are often correlated, older black Americans and older white Americans tend to have similar levels of mental health.
The physical health gap between white Americans and black Americans does not increase in later life, but instead remains consistent or declines. In fact, there appears to be a crossover in health after the age of eighty-five, whereby older black persons gain a slight health advantage.
The most reliable data for this crossover effect come from mortality statistics. The ratio of black men and women to white men and women who die in each ten-year age group declines steadily from age twenty-five onward. American data from 1992 show that the black to white mortality ratio for twenty-five-year-old to thirty-five-year-old males is 2.39, and for females it is 2.72. These ratios decline for each successive age group until a crossover occurs in the eighty-five and older age group. In this age group the ratio is .99 for women and .95 for men, suggesting that among the oldest old, black men and women have a slight advantage over white men and women regarding mortality.
Selectivity in survival is the most common explanation given for the age-based decline in the racial health gap and the crossover effect in mortality ratios. This selectivity explanation suggests that the reason the gap in health status between older white and black adults remains the same or declines in later life has to do with the fact that black Americans are at greater risk of dying than are white Americans at each life-course stage. Hence, only the healthiest black persons live into old age, thereby reducing the health distinction between white and black Americans in later life.
To the extent that status, power, housing conditions, and health reflect overall well-being, the information presented above suggests that, all things considered, older adults tend to have lower levels of well-being than younger adults. However, the relationship between age and well-being is very complex. Indeed, there is notable diversity among older adults, with class, age, gender, and ethnicity/race intersecting to structure inequality in later life. The first letter of each of these bases of diversity forms the word CAGE. A lot of imagery comes to mind when one thinks of the word cage, and for certain groups of people this imagery is quite accurate. However, one must also recognize that there also is a great deal of variation in inequality. Not all working class, racial, or ethnic minority women suffer from low levels of well-being, even though they are at heightened risk. Hence, a task of future gerontological work on inequality is to assess what contributes to such variation and how individuals make choices that either amplify or diminish the effect of structural circumstances on inequality.
Julie Ann McMullin
See also Economic Well-Being; Generational Equity; Health, Social Factors; Housing.
Belgrave, L. L.; Wykle, M. L.; and Choi, J. M. "Health, Double Jeopardy, and Culture: The Use of Institutionalization by African-Americans." The Gerontologist 33 (1993): 379–385.
Calasanti, T. "Incorporating Diversity: Meaning, Levels of Research, and Implications for Theory." The Gerontologist 36 (1996): 147–164.
Dannefer, D., and Sell, R. R. "Age Structure, the Life Course and 'Aged Heterogeneity': Prospects for Research and Theory." Comprehensive Gerontology B 2 (1988): 1–10.
Fry, C. L. "Age, Aging, and Culture." In Handbook of Aging and the Social Sciences, 4th ed. Edited by R. H. Binstock and L. K. George. New York: Academic Press, 1996. Pages 117–136.
George, L. K. "Social Factors and Illness." In Handbook of Aging and the Social Sciences, 4th ed. Edited by R. H. Binstock and L. K. George. New York: Academic Press, 1996. Pages 229–252.
Golant, S. M., and La Greca, A. J. "The Relative Deprivation of the U.S. Elderly Households as Judged by Their Housing Problems." Journals of Gerontology, Series B, 50B, (1995): S13–S23.
Grabb, E. G. Theories of Social Inequality: Classical and Contemporary Perspectives, 3d ed. Toronto, Canada: Harcourt Brace & Company, 1997.
Krause, N. "Mental Disorder in Late Life: Exploring the Influence of Stress and Socioeconomic Status." In Handbook of the Sociology of Mental Health. Edited by C. S. Aneshensel and J. C. Phelan. New York: Kluwer Academic/Plenum Publishers, 1999. Pages 183–208.
Markides, K. S., and Black, S. A. "Race, Ethnicity, and Aging: The Impact of Inequality." In Handbook of Aging and the Social Sciences, 4th ed. Edited by R. H. Binstock and L. K. George. New York: Academic Press, 1996. Pages 153–170.
McMullin, J. A. "Diversity and the State of Sociological Aging Theory." The Gerontologist 40 (2000): 517–530.
McMullin, J. A. "Theorizing Age and Gender Relations." In Connecting Gender and Ageing: A Sociological Approach. Edited by S. Arber and J. Ginn. Buckingham, England: Open University Press, 1995. Pages 30–41.
McMullin, J. A., and Marshall, V. W. "Ageism, Age Relations, and Garment Industry Work in Montreal." The Gerontologist 41 (2001): 111–122.
McMullin, J. A., and Marshall, V. W. "Structure and Agency in the Retirement Process: A Case Study of Montreal Garment Workers." In The Self and Society in Aging Processes. Edited by C. Ryff and V. W. Marshall. New York: Springer, 1999. Pages 305–338.
Moen, P. "Gender, Age, and the Life Course." In Handbook of Aging and the Social Sciences, 4th ed. Edited by R. H. Binstock and L. K. George. New York: Academic Press, 1996. Pages 171–187.
Mutcher, J. E., and Burr, J. A. "Racial Differences in Health and Health Care Service Utilization in Later Life: The Effect of Socioeconomic Status." Journal of Health and Social Behavior 32 (1991): 342–356.
O'Rand, A. "The Cumulative Stratification of the Life Course." In Handbook of Aging and the Social Sciences, 4th ed. Edited by R. H. Binstock and L. K. George. New York: Academic Press, 1996. Pages 188–207.
O'Rand, A. "The Precious and the Precocious: Understanding Cumulative Disadvantage and Cumulative Advantage Over the Life Course." The Gerontologist 36 (1996): 230–238.
Pampel, F. C. Aging, Social Inequality, and Public Policy. Thousand Oaks, Calif.: Pine Forge Press, 1998.
Pynoos, J., and Golant, S. "Housing and Living Arrangements for the Elderly." In Handbook of Aging and the Social Sciences, 4th ed. Edited by R. H. Binstock and L. K. George. New York: Academic Press, 1996. Pages 302–324.
Turner, B. S. Status. Minneapolis: University of Minnesota Press, 1988.
Weber, M. Economy and Society. New York: Bedminster Press. 1922; reprinted in 1968.
World Health Organization. "Definition of Health." World Wide Web document. www.who.int/
See Consumer price index and COLAS
In mathematics, an inequality is a statement about the relative order of members of a set. For instance, if S is the set of positive integers, and the symbol & is taken to mean less than, then the statement 5 & 6 (read 5 is less than 6) is a true statement about the relative order of 5 and 6 within the set of positive integers. The comparison that is symbolized by & is said to define an ordering relation on the set of positive integers. An inequality is often used for defining a subset of an ordered set. The subset is also the solution set of the inequality. There are
many famous inequalities in the field of mathematics. One of them is the triangle inequality, which states that the length of any side of a triangle must be less than the sum of the other two sides, but greater than the difference between those two sides.
A set is ordered if its members obey three simple rules. First, an ordering relation such as less than (<) must apply to every member of the set; that is, for any two members of the set, call them a and b, either a < b or b < a. Second, no member of the set can have more than one position within the ordering; in other words, a < a has no meaning. Third, the ordering must be transitive; that is, for any three members of the set, call them a, b, and c, if a < b, and b < c, then a < c. There are many examples of ordered sets. The alphabet, for instance, is an ordered set whose members are letters. An encyclopedia is an ordered set whose members are entries that are ordered alphabetically. The real numbers and subsets of the real numbers are also ordered. Consequently, any set that is ordered can be associated on a one-to-one basis with the real numbers, or one of its subsets. The algebra of inequalities, then, is applicable to any set regardless of whether its members are numbers, letters, people, dogs or whatever, as long as the set is ordered.
Inequalities involving real numbers are particularly important. There are four types of inequalities, or ordering relations, which are important when dealing with real numbers. They are (together with their symbols): less than (<), less than or equal to (<), greater than (>), and greater than or equal to (>). In each case, the symbol points to the lesser of the two expressions being compared. Since, by convention, mathematical expressions and statements are read from left to right, the statement x + 2 < 6 is read: x plus two is less than six, while 6 > x + 2 is read: six is greater than x plus two. Algebraically, inequalities are manipulated in the same way that equalities (equations) are manipulated, although most rules are slightly different.
The rule for addition is the same for inequalities as it is for equations:
for any three mathematical expressions, call them A, B, and C, if A > B then, A + C < B + C.
That is, the truth of an inequality does not change when the same quantity is added to both sides of the inequality. This rule also holds for subtraction because subtraction is defined as being addition of the opposite or negative of a quantity.
The multiplication rule for inequalities, however, is different from the rule for equations. It is: for any three mathematical expressions, call them A, B, and C, if A < B, andC is positive, thenAC > BC, but if A < B, and C is negative, then AC < BC.
This rule also holds for division, since division is defined in terms of multiplication by the inverse.
As stated previously, an inequality can be a statement about the general location of a member within an ordered set, or it can be interpreted as defining a solution set or relation. For example, consider the compound expression 5 < x < 6 (read: 5 is less than x and x is less than 6) where x is a real number. This expression is a statement about the general location of x within the set of real numbers. Associating each of the real numbers with a point on a line (called the real number line) provides a way of picturing this location relative to all the other real numbers (Figure 1).
In addition, this same expression defines a solution set, or subset of the set of real numbers, namely all values of x for which the expression is true. More generally, an expression in two variables, such as y > 5x + 6, defines a solution set (or relation) whose members are ordered pairs of real numbers. Associating each ordered pair of real numbers with points in a plane (called the Cartesian coordinate system) it is possible to picture the solution set as being that portion of the plane that makes the expression true (Figure 2).
See also Cartesian coordinate plane.
Ordering relation —An ordering relation is a rule for comparing the members of a set in a way that provides a method for placing each member in a specific order relative to the other members of the set. The integers, and the alphabet are examples of ordered sets.
Relation —A relation between two sets X and Y is a subset of all possible ordered pairs (x,y) for which there exists a specific relationship between each x and y.
Set —A set is a collection of things called members or elements of the set. In mathematics, the members of a set will often be numbers.
Solution set —The solution that subset of an set of an inequality is set which makes the ordered inequality a true statement.
Bittinger, Marvin L, and Davic Ellenbogen. Intermediate Algebra: Concepts and Applications. 6th ed. Reading, MA: Addison-Wesley Publishing, 2001.
Burton, David M. The History of Mathematics : An Introduction. New York: McGraw-Hill, 2007.
Lorenz, Falko. Algebra. New York: Springer, 2006.
Setek, William M. Fundamentals of Mathematics. Upper Saddle River, NJ: Pearson Prentice Hall, 2005.
In mathematics , an inequality is a statement about the relative order of members of a set. For instance, if S
is the set of positive integers , and the symbol < is taken to mean less than, then the statement 5 < 6 (read "5 is less than 6") is a true statement about the relative order of 5 and 6 within the set of positive integers. The comparison that is symbolized by < is said to define an ordering relation on the set of positive integers. An inequality is often used for defining a subset of an ordered set. The subset is also the solution set of the inequality.
A set is ordered if its members obey three simple rules. First, an ordering relation such as "less than" (<) must apply to every member of the set, that is, for any two members of the set, call them a and b, either a < b or b < a. Second, no member of the set can have more than one position within the ordering, in other words, a < a has no meaning. Third, the ordering must be transitive , that is, for any three members of the set, call them a, b, and c, if a < b, and b < c, then a < c. There are many examples of ordered sets. The alphabet, for instance, is an ordered set whose members are letters. An encyclopedia is an ordered set whose members are entries that are ordered alphabetically. The real numbers and subsets of the real numbers are also ordered. As a consequence, any set that is ordered can be associated on a one-to-one basis with the real numbers, or one of its subsets. The algebra of inequalities, then, is applicable to any set regardless of whether its members are numbers, letters, people, dogs or whatever, as long as the set is ordered.
Algebra of inequalities
Inequalities involving real numbers are particularly important. There are four types of inequalities, or ordering relations, that are important when dealing with real numbers. They are (together with their symbols) "less than" (<), "less than or equal to" (≤), "greater than" (>), and "greater than or equal to" (≥). In each case the symbol points to the lesser of the two expressions being compared. Since, by convention, mathematical expressions and statements are read from left to right, the statement x + 2 < 6 is read "x plus two is less than six," while 6 > x + 2 is read "six is greater than x plus two." Algebraically, inequalities are manipulated in the same way
that equalities (equations) are manipulated, although most rules are slightly different.
The rule for addition is the same for inequalities as it is for equations:
for any three mathematical expressions, call them A, B, and C, if A > B then, A + C > B + C.
That is, the truth of an inequality does not change when the same quantity is added to both sides of the inequality. This rule also holds for subtraction because subtraction is defined as being addition of the opposite or negative of a quantity.
The multiplication rule for inequalities, however, is different from the rule for equations. It is: for any three mathematical expressions, call them A, B, and C, if A < B, and C is positive, then AC < BC, but if A < B, and C is negative, then AC > BC.
This rule also holds for division , since division is defined in terms of multiplication by the inverse.
As stated previously, an inequality can be a statement about the general location of a member within an ordered set, or it can be interpreted as defining a solution set or relation. For example, consider the compound expression 5 < x < 6 (read "5 is less than x, and x is less than 6") where x is a real number. This expression is a statement about the general location of x within the set of real numbers. Associating each of the real numbers with a point on a line (called the real number line) provides a way of picturing this location relative to all the other real numbers.
In addition, this same expression defines a solution set, or subset of the set of real numbers, namely all values of x for which the expression is true. More generally, an expression in two variables, such as y > 5x + 6, defines a solution set (or relation) whose members are ordered pairs of real numbers. Associating each ordered pair of real numbers with points in a plane (called the Cartesian coordinate system) it is possible to picture the solution set as being that portion of the plane that makes the expression true.
See also Cartesian coordinate plane.
Bittinger, Marvin L., and Davic Ellenbogen. Intermediate Algebra: Concepts and Applications. 6th ed. Reading, MA: Addison-Wesley Publishing, 2001.
Davison, David M., Marsha Landau, Leah McCracken, and Linda Thompson. Prentice Hall Pre-Algebra. Needham, MA: Prentice Hall, 1992.
McKeague, Charles P. Intermediate Algebra. Fort Worth, TX: Saunders College Publishing, 1995.
KEY TERMS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
- Ordering relation
—An ordering relation is a rule for comparing the members of a set in a way that provides a method for placing each member in a specific order relative to the other members of the set. The integers, and the alphabet are examples of ordered sets.
—A relation between two sets X and Y is a subset of all possible ordered pairs (x,y) for which there exists a specific relationship between each x and y.
—A set is a collection of things called members or elements of the set. In mathematics, the members of a set will often be numbers.
- Solution set
—The solution set of an inequality is that subset of an ordered set which makes the inequality a true statement.
Taking power as being distributed along the dimensions of material reward, and differing life-chances as determined by the market, status position, and access to political influence, then according to liberals such as Friedrich Hayek, inequality is the price to be paid for the dynamic economic growth that is characteristic of capitalism. The societies of real socialism (the then actually existing communist states of the Soviet Bloc), committed as they were to the historicism of the class struggle, sought to ameliorate if not abolish these inequalities, but in fact merely generated novel forms of their own, which were in turn less productive of economic growth and social welfare, and subsequently collapsed under the weight of social discontent. However, the arguments propounded by functionalists provide a rationale for inequality, but not (as is sometimes claimed) a proof of its universality and inevitability. In fact, many of the functionalist tenets may now have to face up to a form of egalitarianism which is no longer hampered by the odium of its communistic connotations. It will not be unstated competition between socio-economic systems which defines the agenda as regards inequality, but rather an investigation of what inequalities are justifiable on their own terms, rather than in comparison to some utopian-based alternative. The realities of social-class-determined inequalities of educational achievement, morbidity and mortality rates, and more generally of social mobility, will have to confront the growing problems of the underclass, of generational inequalities, and inequalities produced by the globalization of capitalism, all of which will be seen as part of the social consequences of the ‘peace dividend’. As societies in the post-Cold-War era come to be graded along the criteria of the political democratic audit, so also will the ‘quality of life’ scale be applied both internally and externally, and the extent and nature of inequality will be scrutinized.
Currently, the existence of inequality, its causes and consequences, particularly as they relate to social class, gender, ethnicity, and locality, continues to occupy the sociological foreground. See also FUNCTIONAL THEORY OF STRATIFICATION; INCOME DISTRIBUTION; JUSTICE, SOCIAL; STRATIFICATION.
in·e·qual·i·ty / ˌiniˈkwälitē/ • n. (pl. -ties) difference in size, degree, circumstances, etc.; lack of equality: social inequality | the widening inequalities in income. ∎ archaic lack of smoothness or regularity in a surface: the inequality of the ground hindered their footing. ∎ Math. the relation between two expressions that are not equal, employing a sign such as ≠ “not equal to,” > “greater than,” or < “less than.” ∎ Math. a symbolic expression of the fact that two quantities are not equal.
The inequalities defined on the integers usually include
< (less than)
← (less than or equal to)
> (greater than)
≥ (greater than or equal to)
≠ (not equal to)
A similar set of inequalities is usually defined on the real numbers; such inequalities can produce errors when used in programming languages because of the inherent inaccuracies in the way real numbers are usually represented.
The term inequality is often applied to any comparison involving algebraic expressions and using the above symbols. A special case is the triangle inequality: |a + b| ← |a| + |b|
where | | denotes the absolute value function.