Linear Equations & Inequalities

Linear equations are the easiest algebra on the AMC, which is exactly why they cause avoidable point losses: under time pressure, people rush and flip a sign. This lesson is short on purpose — the goal is precision, not new ideas.

Solving linear equations

Any equation of the form $ax + b = 0$ with $a \neq 0$ has exactly one solution:

Most AMC linear equations are just this idea wrapped in extra algebra — clear denominators first, expand, collect like terms on one side, then divide.

Inequalities and interval notation

Solving a linear inequality works exactly like solving an equation, with one rule that catches people constantly: multiplying or dividing both sides by a negative number flips the inequality sign.

Solutions are usually reported as intervals. A round bracket/parenthesis means the endpoint is excluded (strict inequality); a square bracket means it's included:

• $x > a$ is the interval $(a, \infty)$
• $x \leq a$ is the interval $(-\infty, a]$
• $a < x \leq b$ is the interval $(a, b]$

A compound inequality (two inequalities chained together, or an “and”/“or” combination of two separate ones) is solved by handling each piece separately and then intersecting (for “and”) or taking the union (for “or”) of the resulting intervals.

Practice problems