Solving Linear Equations & Inequalities

The plan for any linear equation

First, simplify each side on its own: distribute any parentheses and combine like terms. Second, collect the variable terms on one side of the equation — add or subtract a variable term from both sides so the variable appears only once. Third, use opposite operations to finish: the opposite of adding $5$ is subtracting $5$, and the opposite of multiplying by $3$ is dividing by $3$.

Whatever you do to one side, you do to the other. An equation is a balance, and every legal move keeps it balanced.

Fraction coefficients

When the coefficient is a fraction, like $\dfrac{2}{3}x = 16$, don't divide — multiply both sides by the reciprocal. Multiplying by $\dfrac{3}{2}$ turns $\dfrac{2}{3}x$ into $1x$ in a single move: $x = 16 \cdot \dfrac{3}{2} = 24$. One step, no messy division by a fraction.

Inequalities: the sign flip

Solve an inequality exactly like an equation. The only new rule: if you multiply or divide both sides by a negative number, the inequality symbol flips. So $-2x \leq 12$ becomes $x \geq -6$ after dividing by $-2$.

Adding or subtracting never flips the symbol — only multiplying or dividing by a negative does. If you're unsure, test a number from your answer in the original inequality and see if it works.

Worked examples

Try one yourself

Common questions