 Top (left): a 2009 time derivative dt version of the second law showing that positive values of entropy change dS will accrue in all natural cyclical processes (dS > 0) at temperatures above absolute zero (at which point dS=0) quantified by the inequality notation that the time derivative of entropy change is greater than or equal to (≥) zero; as tattooed on the upper back of a high school physics teacher named Alison.  Right: The precursor formulation of the Clausius inequality (1856), which states that for all natural processes the value of N will be greater than zero (N > 0) meaning that uncompensated transformations will accrue in the system at the end of one heat cycle, on the arm of a newly minted 2010 philosophy graduate student.  Bottom (left): the circle integral version of the Clausius inequality between a new and burnt match, indicative of the meaning that entropy is the arrow of time.
In mathematics, an inequality refers to a comparison of two variables or formulas that are not are not equal in a numerical sense.

Overview
The six main inequalities used in thermodynamics are as follows:

a < b | a is less than b.
a ≤ b | a is less than or equal to b.
a ≪ b | a is much less than b.

a > b | a is greater than b.
a ≥ b | a is greater than or equal to b.
a ≫ b | a is much greater than or equal to b.

History
The inequality was first expressed verbally: such as aequales, aequantur, esgale, faciunt, ghelijck, or gleich, and sometimes by the abbreviated form aeq.  The mathematical symbols < and > were first employed by English astronomer-mathematician Thomas Harriot in his posthumous 1631 The Analytical Arts Applied to Solving Algebraic Equations, in which he stated: 

“The mark of the majority (signum majoritatis) as a > b, signifies a greater than b and the mark of the minority (signum minoritatis) to a < b signifies a lesser than b.”

The double bar style of the less than or equal to $\leqq \,$and greater than or equal to $\geqq \,$signs were first employed by French mathematician and geophysicist Pierre Bouger in 1734.  In 1670, a similar reduced single bar notation scheme was employed by English mathematician John Wallis who used single horizontal bar above rather than below the inequality signs and . 

Clausius inequality
The famed "inequality" of the Clausius inequality, the core of the second law of thermodynamics, was introduced at first only verbally in 1856 by German physicist Rudolf Clausius who derived the following equality based (=) equation to hold good in every cyclical process: $\int \frac{dQ}{T} = - N \,$

“In non reversible processes, those in which uncompensated transformations necessarily arise, the magnitude of N has consequently a determinable and necessarily positive value.”

where N is the sum of the equivalence-values involved in the cyclical process. Thus, in mathematical notation, although Clausius, in this year, did not use the inequality (something he began doing in 1862), this would be: $\left\vert N \right\vert > 0 \,$

and we would have the following version of the Clausius inequality (the version defined by Clausius in 1865): $\int \frac{dQ}{T} \le 0 \,$

In any event, this seemingly innocuous mention that "the magnitude of N has a positive value" introduced the notion of irreversibility into Newtonian mechanics and the arrow of time into cosmology, thus bring about the thermodynamic revolution and thus the quantum revolution, the latter being a precipitate of the former, and likely several revolutions yet to come.

d Gibbs inequalities
In 1873, American engineer Willard Gibbs began to apply Clausius' inequality methods to the graphical thermodynamic analysis of the behavior of fluids and began to formulate new thermodynamic "force functions" by taking the slopes of various thermodynamic curves, similar to how Isaac Newton calculated the "force" acting on a moving body as the product of the mass of the body by the time derivative (or slope) of velocity change: $\mathbf{F} = \frac{\mathrm{d}}{\mathrm{d}t}(m \mathbf{v})$

In 1876, Gibbs then parlayed this graphical basis into what he called the "general theory of thermodynamic equilibrium" and in doing so formulated five new versions of partial derivative inequalities, in which certain variables are held constant and thus extended the Clausius inequality to more general varieties of systems, such as chemical systems, open systems, electrical systems, etc. The five central Gibbs inequalities are as follows:

 Inequality(1876 notation) Inequality (modern notation) Description $(\delta \eta)_{\epsilon} \leqq 0 \,$ $dS_U \le 0 \,$ For the equilibrium of any isolated system it is necessary and sufficient that in all possible variations δ of the state of the system which do not alter its energy ε, the variation of its entropy η shall either vanish or be negative. $(\delta \epsilon)_{\eta} \geqq 0 \,$ $dU_S \ge 0 \,$ For the equilibrium of any isolated system it is necessary and sufficient that in all possible variations δ of the state of the system which do not alter its entropy η, the variation of its energy ε shall either vanish or be positive. $(\delta \psi)_{t} \geqq 0 \,$ $dF_T \ge 0 \,$where( $F = U - TS \,$) Where by letting psi ψ represent the available energy [free energy] of the system, expressed by the formula ψ = ε – tη, it is found that for the equilibrium of any isolated system it is necessary and sufficient that in all possible variations δ of the state of the system which do not alter its temperature t, the variation of its available energy ψ shall either vanish or be positive. The difference in the two values of available energy ψ for two different states of the system which have the same temperature represents the work which would have to be expended in bringing the system from one state to another by a reversible process and without change of temperature. $\delta \epsilon \geqq 0 \,$ $dU \ge 0 \,$ The reduced form of the second of equilibrium criterion for systems incapable of thermal changes, whereby we may regard the entropy as having the constant value of zero. $\delta \psi \geqq 0 \,$ $dF \ge 0 \,$ The reduced form of the third equilibrium criterion for systems incapable of thermal changes, whereby we may regard the entropy as having the constant value of zero.

Gibbs goes on to state that the "suffix letter" (subscript) in each case indicates that the quantity which it represents remains constant. Lastly, and most importantly, Gibbs outlines the basics of the so-called thermodynamic "driving forces" of chemical processes:

“The quantities ‘–ε’ (negative energy) and ‘–ψ’ (negative available energy), related to a system without sensible motion, may be regarded as a kind of force-function for the system.”

The first of these ‘–ε’, Gibbs says, is the force-function for constant entropy, i.e. when the states of the systems under consideration have constant entropy. The second of these ‘–ψ’, he says, is the force-function for constant temperature, i.e. when the states of the system under consideration have constant temperature. The etymology of this term "force function" was addressed by Clausius in his 1875 discussion of the symbol U, which seems to have originated in the 1835 work of Irish mathematician William Hamilton; itself tracing to French mathematician Joseph Lagrange’s 1788 ‘central function’, from his Analytical Mechanics.

Lewis inequality
See main: Lewis inequality
In 1923, American physical chemist Gilbert Lewis, starting with the foundation of Gibbs inequalities, which he says are applicable to "chemical processes which are in some way harnessed for the production of useful work", introduced a new inequality as follows:

 Inequality (1923 notation) Inequality (modern notation) Description where( $F = E + PV - TS \,$) where( $G = U + PV - TS \,$) Whereby "no actual isothermal process is possible unless it meets this condition", which applies to "the far more common case of a reaction which runs freely, like the combustion of a fuel, or the action of an acid upon a metal; or, in other words, a system which is subject to no external forces, except a constant pressure exerted by the environment."

In modern format, assuming that from hence forth we use the term "process" to mean all natural isothermal-isobaric processes that occur on the surface of the earth, such as human chemical reactions occurring between people, what Lewis states here is that:

“No actual natural process is possible unless it meets this inequality ΔG < 0 criterion.”

This is huge statement. Getting directly to the point: any hypothetical relationships will only be possible if the free energy change for the process is less than zero. Next, Lewis goes on to state, similar to Gibbs' description of the two force-functions, that:

“We may think of the quantity ‘– ΔF ’ as the driving force of a reaction; where, in a thermodynamic sense, a system is stable when no process can occur with a diminution in free energy.”

This, again, is a huge statement. Lewis introduce here not only the notion that free energy change is the driving force of earth-bound natural processes, as was described in 1718 by Isaac Newton in terms of respective values of the force of chemical affinity A as being the driving force behind reactions:

 Newton notation (1718) Lewis notation (1923) Partington notation (1924) Modern notation (1933) $A > 0 \,$ $A = - \Delta F \,$ $A = -\Delta Z \,$ $A = - \Delta G \,$ The position of G1 is such that, in the words of Gilbert Lewis (1923), "no further process can occur with a diminution in free energy", and is thus representative of a state of maximal stability; whereas the position of G1 could decrease further in free energy, to the position of state one, and is thus not maximally stable.

but also introduces the notion of "stability", i.e. that when the variation of free energy for the process or reaction reaches its lowest value, or, as Lewis puts it, when "no further process can occur with a diminution in free energy", the system will be said to be stable, as would be indicated graphically by the lowest point of free energy on the reaction coordinate, as depicted adjacent.

In 1933, English physical chemist Edward Guggenheim, elaborated on this logic as well, and was the one who assigned the symbol G and the name Gibbs free energy, to the isothermal-isobaric inequality for natural processes with occur freely on the surface of the earth, after which it is said that modern chemical thermodynamics was born.

Lipmann inequality
In 1941, in his studies of the energetics of frog leg muscle contractions, German-born American biochemist Fritz Lipmann introduced the concept of “coupling” into the inequality, in the sense that weakly non-spontaneous reactions (ΔG > 0) can be coupled energetically to strongly spontaneous reactions (ΔG << 0), in such a manner that the latter can drive the former. In short, using the rule that the various free energy changes for the reactions in a given system can be summed to yield the total system free energy change: $\Delta G_{total} = \sum_{i=1}^k \Delta G_i$

Lipmann postulated that as long the magnitude of the free energy changes for the exergonic reactions is greater than that of the sum of the free energy changes for the endergonic reactions, that free energy released from the more powerful reactions will drive the weaker free energy absorbing reactions. Thus, for example, if one spontaneous reaction (A → B)SR is mixed with a non-spontaneous reaction (C → D)NR, such as shown below, then the total system process will proceed, the spontaneous reaction driving the non-spontaneous reaction, as long as: |ΔGAB| > |ΔGCD|, or in general terms as long as: $\sum_{i=1}^k \left | G_{SR} \right \vert_i > \sum_{i=1}^k \left | G_{NR} \right \vert_i$

In the decades to following, this "coupling" theory has found nearly universal acceptance and application in the explanation of internal biochemical processes. The extrapolation as to how this applied to coupling in social terms is a nascent subject in infancy, in attempting to explain prevalent seemingly detrimental phenomenon such as nepotism as well as the phenomena of the "multigenerational push", as evidenced in the seven-generation Ivy league pressurized creation of American engineer Willard Gibbs' 1876 Equilibrium, written quickly in a period of three years, during which time, as he says, “I had no sense of the value of time, of my own or others, when I wrote it”, and as exemplified by American economist Paul Samuelson’s famed 1947 Nobel Prize winning PhD dissertation-turned-book Foundations of Economic Analysis that he wrote quickly in the pace of one year, because, as he says, “I was a young man in a hurry, because I felt the limited lifetime of may male ancestors tolled the knell for me.” 

Gibbs-Bogolyubov inequality
The term Gibbs-Bogolyubov inequality refers to the work of Russian mathematical physicist Nikolay Bogolyubov who sometime towards the mid 20th century introduced a quantum mechanical version of one of Gibbs inequalities. The Gibbs-Bogolyubov inequality, supposedly, takes something of the form of: $F_2 - F_1 \ge \left \langle U_2 - U_1 \right \rangle_2 \,$

which relates the Helmholtz free energies F of two systems that share a common phase space and are at the same temperature T, but whose Hamiltonians differ by U2 – U1; the angle brackets signifying "average value". A tabulated view of the nature of the greater than > or less than < zero inequality aspects of the standard Gibbs free energy change ΔG° and equilibrium constant Keq quantification methods of reactions, expressed by the van’t Hoff equation, ΔG° = – RT ln Keq, for a generic reversible reaction, e.g. A + B ⇌ C + D, where a large positive equilibrium constant (ΔG ≪ 0) signifies a reaction that goes strongly, completely, and spontaneously in the forward direction towards the formation of products. 

Chemical reactions
At this point, we interject to answer the age old question: why do some reactions take place on their own accord while others do not? The inequality sign is the physical chemistry answer to this question. The answer is that reactions that take place on their own have a free energy change (ΔG or ΔF) that is less than ‘<’ zero, a principle that was established, based on the Gibbs inequality (1876), which itself was based the Clausius inequality (1856) in the 1882 paper "On the Thermodynamics of Chemical Processes" by German physicist Hermann Helmholtz.

Spontaneity criterion
See main: Spontaneity criterion
The much greater than (a ≪ b) or much less than (a ≪ b) notations are sometimes used to depict levels of spontaneity in the quantification of chemical reactions, according to the following rule:

● The notation ΔG ≪ 0 (as compared to ΔG < 0) means that the free energy change for the process is much less than zero and will thus be greatly spontaneous.
● The notation ΔG ≫ 0 (as compared to ΔG > 0) means that the free energy change for the process is much greater than zero and will thus be greatly non-spontaneous.

This much greater than and much less than notation is thus greatly relevant to the processes involved in actions of human existence, such as in falling in love or spontaneously completing a masterpiece, both of which being processes quantified by ΔG ≪ 0, which contrasts to relationship states or destruction or weakly happening processes that may be near the break-up point ΔG < 0, at the breakup point ΔG = 0, past the break up point ΔG > 0, or much past the break up point ΔG ≫ 0.

This can be expressed in terms of an equilibrium constant, or the ratio of the concentrations of products to reactants. In a reactive system of 100 single men A and a 100 single women B, each sex assumed to be the same generic type of molecule, for instance, the pairing reaction would have the form:

A + B ⇌ AB

and the equilibrium constant for this reaction would be: $K_{eq}=\frac{[AB]}{[A][B]} \,$

Hence, if the value of ΔG° is negative (Keq positive), termed an exergonic reaction, then the reaction will proceed from left to right, and the more negative the value of ΔG°, the further towards the right will the reaction proceed. If the value of ΔG° is positive (Keq negative), termed an endergonic reaction, the reaction proceeds in the reverse direction as written or from right to left. Sodium chloride NaCl put in contact with water H20 after which a solvation-type dissolution reaction occurs.
Sodium spontaneity
A classic example to help visualize the nature of the inequality differences between the two types of free energy changes, i.e. those greater than zero as compared to those less than zero, is the difference between the reactions that occur when (a) sodium chloride NaCl, otherwise known as table salt, is put in contact with water H2O as compared to (b) when sodium Na is put in contact with water H2O.

In the first scenario, a lightly energetic dissolution reaction will occur owing to the fact that the ionic bonds that hold Na to Cl are weak and will easily break when NaCl is put in contact with water. The reaction is not favored enthalpically (ΔH=4kJ/mol), but is favored entropically, i.e. favored not in a heat sense, but yest in an ordering sense.

To explain this, the solvation process will be thermodynamically favored only if the overall Gibbs energy of the solution is decreased (ΔG < 0), compared to the Gibbs energy of the separated solvent and solid (or gas or liquid). This means that the change in enthalpy minus the change in entropy (multiplied by the absolute temperature) is a negative value, or that the Gibbs free energy of the system decreases. Enthalpy of solvation can help explain why solvation occurs with some ionic lattices but not with others. The difference in energy between that which is necessary to release an ion from its lattice and the energy given off when it combines with a solvent molecule is called the enthalpy change of solution. A negative value for the enthalpy change of solution corresponds to an ion that is likely to dissolve, whereas a high positive value means that solvation will not occur. It is possible, however, that an ion will dissolve even if it has a positive enthalpy value (ΔH=3.88 kJ/mol, for sodium chloride solvation), whereby the extra energy required comes from the increase in entropy that results when the ion dissolves (ΔS > 0). 

In the second scenario, a powerful volume-expanding explosive reaction will occur in which a mixture of solid and liquid is transformed into a mixture of liquid and gas, during which process strong covalent bonds are broken in the water molecule by the strong affinity of sodium for oxygen, as a result of some of the hydrogen atoms are displaced from their former attachment to the oxygen to form hydrogen gas H2(g).

 2Na(s) + 2H20 (l) → 2NaOH(aq) + H2(g) 0 -237 kJ/mol -418k kJ/mol 0

By looking up the standard free energies of formation for each of these entities, in this case Helmholtz free energies of formation are listed (bottom row), we can then calculate the free energy change for this reaction to determine its predetermined level of energetic spontaneity, which is calculated to be -359 kJ/mol, and thus we say that the reaction is high spontaneous (ΔF ≪ 0).

References
1. (a) Earliest uses of Symbols of Relation (2007) - Jeff560.Tripod.com.
(b) Cajori, Florian. (1928). A History of Mathematical Notations (Vol. I) (pg. 297) (online). Publisher.
2. Harriot, Thomas. (c.1620). The Analytical Arts Applied to Solving Algebraic Equations (Artis Analyticae Praxis ad Aequationes Algebraicas Resolvendas). Publisher, 1631.
3. (a) Ball, W.W. Rouse. (1960). A Short Account of the History of Mathematics. Dover.
(b) Pierre Bouguer – Wikipedia.
4. (a) Cajori, Florian. (1928). A History of Mathematical Notations (Vol. II) (pg. 118) (online). Publisher.
(b) John Wallis – Wikipedia.
5. Clausius, Rudolf. (1865). The Mechanical Theory of Heat (pg. 144). John van Voorst.
6. (a) Zimmerman, Carl. (2009). “I am Shiva, the Physics Teacher of the Worlds”, Discover Science Blogs, Apr. 23.
(b) Entropy generation (tattoo) – full picture.
(c) Quote: “equation describing entropy, symbolizes destruction, simply stating that this fundamental breakdown of systems and accumulation of disorder either increases or stays the same over time, but never decreases.”
10. Perry, John H. (1984). Perry's Chemical Engineers' Handbook. McGrawHill.
11. (a) Solvation – Wikipedia.
(b) Free energy of solvation – Wikipedia.
(c) Enthalpy of solution – Wikipedia.
(d) Lide, David R. (2004). CRC Handbook of Chemistry and Physics. CRC Press.
12. Widom, B. (1990). “Two Ideas form Gibbs: the Entropy Inequality and the Dividing Surface”, in: Proceedings of the Gibbs Symposium: Yale University, May 15-17, 1989 (pgs. 73-). Yale University Press.
13. (a) Cropper, William H. (2004). Great Physicists: the Life and Times of Leading Physicists from Galileo to Hawking, (section II: Thermodynamics, pgs. 41-134; ch. 9: “The Greatest Simplicity: Willard Gibbs”, pgs 106-23). Oxford University Press.
(b) Szenberg, Michael, Gottesman, Aron A. and Ramrattan, Lall. (2005). Paul A. Samuelson: on Being an Economist (Samuelson’s 1940 dissertation, pgs. 20-22). Jorge Pinto Books.
14. Cropper, William H. (2001). Great Physicists: the Life and Times of Leading Physicists from Galileo to Hawking (pgs. 108-109). Oxford University Press.
15. (a) The equation shown uses a modern variation of German physicist Carl Neumann 1875 notation for the inexact differential, of little Greek delta δ.
(b) Equation on arm was taken from the mathematical descriptions section of the Wikipedia on second law of thermodynamics article, as was added by American chemical engineer Libb Thims on 18 Sep 2006.
(c) Zimmer, Carl. (2010). “Graduating into Entropy”, Discover Magazine Blog, May 02.
(d) The Second law of Thermodynamics (05 Apr 2010) – GeekyTattoo.com.