Algebra - Solving linear equations and inequalities

A linear equation is an algebraic equation where each term has an exponent of at most 1. Visually, when plotted on a Cartesian plane, it forms a perfectly straight line, representing a constant rate of change.

The Principle of Equality acts like a balance scale. To maintain the equilibrium of the equation, any operation (addition, subtraction, multiplication, or division) performed on one side must be applied identically to the other side.

The Distributive Property, $a(b + c) = ab + ac$, is essential for removing brackets. This is often the first step in simplifying complex linear equations where variables are grouped inside parentheses.

Solving by isolating the variable involves moving all terms containing the variable (like $x$) to one side of the equals sign and all constant numerical values to the other, using inverse operations to 'undo' the existing operations.

Equations involving fractions can be simplified by multiplying every term by the Lowest Common Denominator (LCD) to 'clear' the denominators, or by using cross-multiplication when the equation is a simple proportion like $\frac{a}{b} = \frac{c}{d}$.

Linear inequalities represent a range of possible values rather than a single solution. On a number line, the solution is visualized as a shaded ray or segment. An open circle $\circ$ indicates the endpoint is not included ($<$ or $>$), while a solid dot $\bullet$ indicates it is included ($\le$ or $\ge$).

The Negative Multiplier Rule for inequalities states that if you multiply or divide both sides of an inequality by a negative number, the inequality sign must be reversed (e.g., from $<$ to $>$). This reflects the reversal of order on the negative side of the number line.

A solution to an equation is the specific value of the variable that makes the statement true. You can always check your work by substituting the value back into the original equation to see if both sides balance.