What is the Difference Between Inequalities and Equalities?

An equality states that two expressions have the same value, while an inequality indicates that two expressions have different values or a specific order relationship.

To understand this difference better, let's break it down:

Equalities: Showing Sameness

• Definition: An equality is a mathematical statement that asserts two expressions have the same value.
• Symbol: Equalities use the equals sign (=).
• Example: x + 2 = 5 means the expression x + 2 has the same value as 5. The solution to this equality is x = 3.
• Solution: An equality typically has a specific solution or set of solutions that make the statement true.

Inequalities: Showing Difference or Order

• Definition: An inequality is a mathematical statement that compares two expressions using inequality symbols.
• Symbols: Common inequality symbols include:
  • < (less than)
  • > (greater than)
  • ≤ (less than or equal to)
  • ≥ (greater than or equal to)
  • ≠ (not equal to)
• Example: x + 2 < 5 means the expression x + 2 is less than 5. The solution to this inequality is x < 3. This means any value of x that is less than 3 will satisfy the inequality.
• Solution: An inequality usually has a range of solutions that make the statement true. The solution set can be represented on a number line or using interval notation.

Table Summarizing the Key Differences

Feature	Equality	Inequality
Definition	Shows that two expressions have the same value	Shows that two expressions have different values or a specific order relationship
Symbol	= (equals)	<, >, ≤, ≥, ≠
Solution	Typically a specific value or set of values	Typically a range of values
Representation	Usually a single point on a number line	Usually an interval or multiple intervals on a number line

Examples to Illustrate the Difference

• Equality: The equation 2x = 6 has one solution: x = 3.
• Inequality: The inequality 2x > 6 has infinitely many solutions: all values of x greater than 3. This can be represented as x > 3.

In summary, the key difference lies in the relationship they express: equalities assert sameness, while inequalities indicate a difference or order. Equalities have specific solutions, while inequalities typically have a range of solutions.