Equations & Inequalities

Equation: A mathematical statement showing two expressions are equal, using an equals sign ($=$). Example: $x + 5 = 12$.

Solution: The value(s) that make the equation true. For $x + 5 = 12$, the solution is $x = 7$ because $7 + 5 = 12$ is true.

Inverse operations: Operations that undo each other: addition $\leftrightarrow$ subtraction, multiplication $\leftrightarrow$ division. To undo $+5$, subtract 5. To undo $\times 3$, divide by 3.

Goal: Isolate the variable on one side of the equation using inverse operations. Whatever you do to one side, you must do to the other side to keep the equation balanced!

Balancing principle: An equation is like a balanced scale. If you add 3 to one side, you must add 3 to the other side to keep it balanced: $x + 5 = 12$ becomes $x + 5 + 3 = 12 + 3$.

Checking solutions: Always substitute your answer back into the original equation to verify it works. For $x = 7$ in $x + 5 = 12$: $7 + 5 = 12$ ✓