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Concept

Equivalent Equation and Equivalent Inequality

These two concepts will be discussed one at a time.

Equivalent Equation

Two equations are called equivalent equations if they have the same solution(s). Equations are often solved by applying the Properties of Equality. Each time a property is applied, an equivalent equation is produced. Consider the following example.
To solve this equation, like terms containing the variable y will be grouped on the left-hand side and constants will be grouped on the right-hand side.
5y+3=3y+15
5y+33y=3y+153y
2y+3=15
2y+33=153
2y=12
y=6
Here, as a result of applying different Properties of Equality, six equations equivalent to the first one were created. They are equivalent as 6 is the solution to every one of the above equations. This observation can be checked by verifying the solution.
Therefore, y=6 is indeed a solution to the original equation. Note that in the process of checking, three more equivalent equations were formed.

Equivalent Inequality

Two or more inequalities are equivalent inequalities if they have the same solution set. Similar to equivalent equations, applying Properties of Inequality to an inequality produces an equivalent inequality. Consider the following example.
To solve this inequality, three Properties of Inequalities will be used.
3x+21>62x
5x+21>6
5x>-15
x>-3
The solution set of every inequality above is x>-3. The Properties of Inequalities produced three equivalent inequalities until the solution set was found.

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