Inequalities represent the comparative evaluation of the variables on the left to those on the right of the ‘<’ or ‘>’ sign. Alternatively, equations represent the equality of the variables on the left and right sides of the ‘=’ sign.

Inequalities compare the relative size of values, while equations prove them to be equal. This seminal difference also gives rise to a host of other differences that must be cognized.

**Inequalities vs Equations**

The difference between inequalities and equations is in terms of their definitions that in turn influence their use in mathematical problems. While inequalities are used to represent the unequal relationship between a set of variables, equations are used to symbolically represent the equality of the two sets of variables used.

**Comparison Table Between Inequalities and Equations**

Parameters of Comparison | Inequalities | Equations |

Definition | It is a mathematical statement that represents the inequality and order of variables on the left and right sides. | It is a mathematical statement that represents the equality between the left and right side variables in an equation. |

Symbols Used | The ‘greater than’ and ‘lesser than’ signs are used to symbolically represent the relationship between variables. | The ‘equal to’ sign is used to symbolically represent the relationship between variables |

Representational Function | Represent inequality between the variables used. | Represent equality between the variables used. |

Solutions | A solution set -with infinite answers- is a plausible result for an inequality. | The solution for an equation is fixed and singular. |

Number of Roots | The total number of roots for inequalities is infinite. | The total number of roots for equations is definite. |

**What are Inequalities?**

Inequalities are mathematical statements that represent the unequal relationship between a set of variables. They employ the ‘>’ or ‘<’ signs to signify the comparative analysis of the variables used. Inequalities necessarily represent the order of the relationship between the variables used.

They are also used in mathematical problems to compare the relative size of values. Inequalities can be presented in two ways.

Their presentation may have a close semblance to equations or they may also be presented as a simple statement of fact- like in mathematical theorems. Inequalities are commonly used to compare integers, variables, and other algebraic expressions.

Some examples of inequalities are:

**‘c > d’, where ‘c’ is greater than ‘d’.**

**‘c < d’, where ‘c’ is lesser than ‘d’.**

There can be a number of variants among inequalities, including strict and compound inequalities. Each of these variants has a given set of rules to determine the resultant solution set.

**What are Equations?**

Equations are also mathematical statements that are used to represent the equality of variables on the left and right sides of the statement. They use the ‘=’ sign to represent the equality of the values of the two given sets of algebraic variables. In an equation, the solution is always unitary and representative of the equality between the left and right sides.

Some examples of equations are:

**a + 2 = 30**, where ‘a + 2’ and ‘30’ are both algebraic expressions, separated by the ‘=’ sign.

**5a + 5 = 35**, where ‘5a + 5’ and ‘35’ are both algebraic expressions, separated by the ‘=’ sign.

Commonly, equations include more than one variable. In the examples given above, the process of solving the equation refers to finding out the value of the unknown variable. Equations are used extensively in algebraic calculations.

Equations can also be of various types like linear and simultaneous equations and quadratic equations.

**Main Differences Between Inequalities and Equations**

- The main difference between inequalities and equations is in terms of their definitions that clearly delineate their functionalities in mathematical operations. An equation -as the name suggests- represents the equality between two variables in the given formulation. The left side of an equation is invariably equal to the right side. Inequalities, on the other hand, are mathematical statements of the inequality between variables. The left and right sides of inequalities represent variables as greater than or less than- highlighting their inequality and relative sizes.
- The second seminal difference between the two is in terms of what they each represent. While inequalities connote the inequality between two variables, equations are used to represent the equality between two variable quantities.
- The symbols used to express equality and inequality in each of these are also different. Inequalities utilize ‘>’ and ‘<’ symbols to represent the inequality between variables, while equations represent equality between given variables by using alphabetical symbols like ‘a’ and ‘b’ accompanied by the mandatory ‘equal to’ sign between the left and right sides. Inequality signs are used in the former, while the signs of equality are implemented in the latter.
- Inequalities and equations are also significantly different in terms of their potential solutions. Multiple answers may be possible for inequalities. A ‘solution set’- comprising of infinite values- is prescribed as a befitting solution for an inequality. On the other hand, only one answer can be determined for an equation.
- Lastly, the total number of roots of an equation are definite. This is not the case for inequalities.

**Conclusion**

Both inequalities and equations are fairly common mathematical statements used to represent the relationship between a set of variables. Although both are solved using similar techniques, there exist seminal differences between the two that need to be cognized.

The most important difference between the two is in terms of the kind of representation each offers to the variables used. While inequalities represent the unequal relationship between the two variables in the mathematical statement, equations represent the equality between the variables.

Both these mathematical statements use different symbols to express the relationship between variables. The former uses the ‘greater than’ and ‘lesser than’ symbols to symbolically represent the unequal association of variables. The latter uses an ‘equal to’ sign to represent the equality of the left and right sides of the equation.

The possible solutions for each are also varied, such that the former may have multiple plausible results while the latter has a definite, singular solution. These differences need to be noted to understand the operation of each of these mathematical forms of representation.

**References**

- https://ieeexplore.ieee.org/abstract/document/754846/
- http://www.mathematik.uni-dortmund.de/~erme/CERME4/CERME4_WG6.pdf#page=24

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