Equations & Inequalities

Linear equations, systems, inequalities, absolute value and word problems.

🔢 Solving Linear Equations

Part 1 of 7 — One-Step, Two-Step, Multi-Step & Variables on Both Sides

Linear equations are the backbone of ACT Algebra. You will see several on every test.

Golden Rule: Whatever you do to one side, do to the other.

📐 Systems of Equations

Part 2 of 7 — Substitution, Elimination & Word Problems

A system of equations is two (or more) equations with the same unknowns.

Goal: Find the $(x, y)$ pair that satisfies both equations simultaneously.

Graphing Inequalities — Number Line Summary

ACT Tip: Open circle = strict ($<, >$). Closed circle = inclusive ($\leq, \geq$).

📝 Linear Word Problems

Part 4 of 7 — Translating Words to Equations, Rate/Distance/Time & Mixtures

The hardest part of word problems is translation — turning English into algebra.

Watch the reversal: "less than" names the starting quantity second in English but you write it first in algebra. "7 less than $3n$" means start at $3n$ and subtract 7. (Order doesn't matter for "more than," since $3n + 7 = 7 + 3n$.)

Consecutive integers:

• Consecutive integers: $n,\ n + 1,\ n + 2$
• Consecutive even integers: $n,\ n + 2,\ n + 4$ ($n$ even)
• Consecutive odd integers: $n,\ n + 2,\ n + 4$ ($n$ odd) — still $+2$, because odd numbers are also 2 apart

Example: three consecutive odd integers sum to 57 → $3n + 6 = 57 \implies n = 17$, so the integers are 17, 19, 21.

Geometry facts word problems assume: a circle's diameter = 2r (the radius is half the diameter); circumference $= 2\pi r = \pi \cdot$ diameter; area $= \pi r^2$. If a problem gives the diameter, halve it before using $\pi r^2$.

ACT Tip: Underline what they're asking for before you set up the equation.

📏 Absolute Value Equations

Part 5 of 7 — Solving $|x - a| = b$, Two Cases & Extraneous Solutions

The absolute value $|A|$ is the distance from $A$ to 0 on the number line. Because distance is always non-negative:

$|A| = b \implies A = b \quad \text{or} \quad A = -b \qquad (b \geq 0)$

Key Insight: Always isolate the absolute value expression first, then split into two cases.

🔧 Algebraic Manipulation

Part 6 of 7 — Factoring, Distributing, Combining Like Terms & Special Products

Strong algebraic manipulation skills save time on every section of the ACT Math. These are the building blocks.

🏆 Review & Mixed Practice

Part 7 of 7 — Cheat Sheet, Mixed ACT-Style Problems & Time Strategy

This final part pulls together everything from Parts 1–6.

Quick-Reference Cheat Sheet