Equations, Expressions, Formulae, and Inequalities

In Algebra, we work with numbers, symbols, and letters to represent values. Let’s dive into what makes up the core concepts in algebra: equations, expressions, formulae, and inequalities. Each has its own job, and understanding them will make solving algebra problems a lot easier

Expression

Definition: An expression is a combination of numbers, variables (like $x$ or $y$), and operations (like $+, -, \times, \div$).

Think of an expression as a “mathematical phrase” that doesn’t have an equals sign (=).

Example: 

• $3x + 5$
• $4a - 7b + 12$

Note: An expression can’t be “solved” because it doesn’t have an equals sign; it only represents a value.

 

Equation

Definition: An equation is like a complete “mathematical sentence.” It shows that two expressions are equal to each other by using an equals sign (=)

Example:

• $2x + 3 = 7$
• $a + 4 = 12$

Equations can be solved to find the value of the unknowns (like $x$ or $a$).

 

Formula

Definition: A formula is an equation that expresses a relationship between different variables. Formulas are used to calculate things, like finding the area or volume.

Example:

• The formula for the area of a rectangle is: $$A = l \times w$$

where:

• $A$ is the area,
• $l$ is the length, and
• $w$ is the width

 

Inequality

Definition: An inequality shows that two values or expressions are not necessarily equal but are instead greater than, less than, greater than or equal to, or less than or equal to each other.

Symbols: 

• $\gt$ means “greater than.”
• $\lt$ means “less than.”
• $\ge$ means “greater than or equal to.”
• $\le$ means “less than or equal to.”

Example:

• $x + 5 \gt 10$
• $3y -7 \le 12$

Solving an Inequality Example: Let’s solve $2x + 3 \lt 11$

1. Subtract 3 from both sides: $$2x \lt 8$$
2. Divide by 2: $$x \lt 4$$