Equations and Inequalities Cheat Sheet

Equations and inequalities are fundamental mathematical tools used to represent and solve problems across various real-world contexts, from science and engineering to finance and everyday life. Understanding how to manipulate and solve them is key to problem-solving.

Core Principles

An equation is a mathematical statement that two expressions are equal, using the '=' sign.
A solution to an equation is a value that makes the equation true.
Equivalent equations are equations that have the same solution(s). They can be obtained by performing the same operation on both sides of an equation.
An inequality is a mathematical statement comparing two expressions using symbols like <, >, ≤, or ≥.
Solving inequalities involves similar operations to solving equations, with the key difference that multiplying or dividing by a negative number reverses the inequality sign.
Formulas and relationships describe how variables are connected, allowing us to calculate unknown values by substituting known values and solving the resulting equation.

Formulas

Area of a rectangle: $A = l \times w$
Perimeter of a rectangle: $P = 2l + 2w$
Simple quadratic equation: $ax^2 = c$
Temperature conversion (Celsius to Fahrenheit): $F = 1.8C + 32$
Area of a trapezium: $A = \frac{1}{2}(a+b)h$