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 other differences that must be cognized.
Key Takeaways
- Equations are mathematical statements that assert the equality of two expressions; inequalities indicate a relationship of greater than, less than, or not equal between two expressions.
- Equations can have a finite number of solutions; inequalities can have infinite solutions.
- Equations represent a specific point or value; inequalities represent a range of values that satisfy the statement.
Inequalities vs. Equations
An equation is a statement that shows the equality between two expressions to find the values of the variables that make the equation true. An inequality is a statement that shows a relationship between two expressions that are not necessarily equal and is used to compare the values of two variables.
Comparison Table
| 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 representing the equality between an equation’s left and right side variables. |
| Symbols Used | The ‘greater than’ and ‘lesser than’ signs symbolically represent the relationship between variables. | The ‘equal to’ sign is used to represent the relationship between variables. symbolically |
| 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 be 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 several 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 representing 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 above, 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, simultaneous, and quadratic equations.
Main Differences Between Inequalities and Equations
- The main difference between inequalities and equations is their definitions that delineate their functionalities in mathematical operations. As the name suggests, an equation 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 what they each represent. While inequalities connote the inequality between two variables, equations 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 equality signs 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 infinite values- is prescribed as a befitting solution for inequality. On the other hand, only one answer can be determined for an equation.
- Lastly, the total number of roots of an equation is definite. This is not the case for inequalities.
- https://ieeexplore.ieee.org/abstract/document/754846/
- http://www.mathematik.uni-dortmund.de/~erme/CERME4/CERME4_WG6.pdf#page=24
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