ACT Math: Algebra Review — Everything You Need to Know

Algebra is the backbone of the ACT Math section. Roughly 40–50% of the 50 questions on the Enhanced ACT involve algebraic thinking in some form — from straightforward equation solving to systems of equations to translating word problems into algebraic expressions.

The good news? ACT algebra isn't trying to stump you with obscure topics. It tests the same core concepts over and over, just wrapped in different-looking problems. Once you learn to identify what type of equation you're dealing with and have a reliable approach for each type, these questions become some of the most predictable points on the entire test.

This guide covers every algebra topic you'll encounter on the ACT, organized by concept. Think of it as the review you wish your math class had given you before test day.

Variables and Equations: The Foundation

Every algebra problem on the ACT comes down to one thing: solving for an unknown value. Whether the question uses x, n, or some other variable, your job is to isolate it.

Solving Linear Equations

A linear equation has the variable raised to the first power — no exponents, no square roots, no fractions with the variable in the denominator. These are the simplest equation type on the ACT and should be automatic.

Example: Solve 3x + 1 = 10

1. 1.Subtract 1 from both sides: 3x = 9
2. 2.Divide both sides by 3: x = 3

That's it. The ACT will dress this up — bigger numbers, more terms, variables on both sides — but the process is always the same:

• •Combine like terms on each side
• •Move variable terms to one side and constants to the other
• •Isolate the variable by dividing

Multi-Step Equations

When equations have variables on both sides, fractions, or distributive property, stay organized:

Example: 2(x + 3) = 5x − 9

1. 1.Distribute: 2x + 6 = 5x − 9
2. 2.Move variables to one side: 6 = 3x − 9
3. 3.Move constants: 15 = 3x
4. 4.Divide: x = 5

Key tip: When the ACT gives you answer choices, you can always plug in the answers and see which one works. This is especially useful when the algebra looks messy. Start with the middle answer choice — if it's too big, go smaller (and vice versa). This strategy alone can save you on the hardest algebra questions.

Inequalities

Inequalities work exactly like equations with one critical exception: when you multiply or divide by a negative number, flip the inequality sign.

Example: −2x + 4 > 10

1. 1.Subtract 4: −2x > 6
2. 2.Divide by −2 (flip the sign!): x < −3

The ACT loves to test whether you remember to flip the sign. It's one of the most common traps on the test.

Polynomial Basics

A polynomial is any expression that combines variables and constants using addition, subtraction, and multiplication — with whole-number exponents only.

Key Vocabulary

• •Monomial: One term (e.g., 5x²)
• •Binomial: Two terms (e.g., x + 3)
• •Trinomial: Three terms (e.g., x² + 5x + 6)
• •Degree: The highest exponent (e.g., x² + 5x + 6 has degree 2 — a quadratic)

You don't need to memorize these vocabulary terms for the ACT, but understanding them helps you quickly classify a problem and select your approach.

Adding and Subtracting Polynomials

Combine like terms — terms with the same variable and exponent:

(3x² + 2x − 5) + (x² − 4x + 1) = 4x² − 2x − 4

Watch the signs carefully when subtracting — distribute the negative:

(3x² + 2x − 5) − (x² − 4x + 1) = 3x² + 2x − 5 − x² + 4x − 1 = 2x² + 6x − 6

Multiplying Polynomials (FOIL and Beyond)

When multiplying two binomials, use FOIL — First, Outer, Inner, Last:

(x + 3)(x − 2) = x² − 2x + 3x − 6 = x² + x − 6

For larger polynomials, multiply every term in the first polynomial by every term in the second, then combine like terms. The ACT rarely asks you to multiply anything larger than two binomials.

Solving Polynomial Equations

This is where algebra gets more interesting — and where more points are available on the ACT.

Factoring Quadratics

Factoring is the most important algebraic skill on the ACT. If you can factor quickly and accurately, you'll pick up points on both easy and hard questions.

Standard form: ax² + bx + c = 0

When a = 1: Find two numbers that multiply to c and add to b.

Example: x² + 5x + 6 = 0

• •What multiplies to 6 and adds to 5? → 2 and 3
• •Factor: (x + 2)(x + 3) = 0
• •Solutions: x = −2 or x = −3

When a ≠ 1: Use the AC method — multiply a × c, find factors that add to b, then rewrite and group.

Example: 2x² + 7x + 3 = 0

• •a × c = 6. What multiplies to 6 and adds to 7? → 1 and 6
• •Rewrite: 2x² + x + 6x + 3 = 0
• •Group: x(2x + 1) + 3(2x + 1) = 0
• •Factor: (x + 3)(2x + 1) = 0
• •Solutions: x = −3 or x = −1/2

The Quadratic Formula

When factoring isn't obvious, the quadratic formula always works:

x = (−b ± √(b² − 4ac)) / 2a

This formula is not provided on the ACT — you need to have it memorized. The good news: if you've memorized it and can plug in carefully, it works 100% of the time on any quadratic.

Pro tip: The expression under the square root — b² − 4ac — is called the discriminant. The ACT sometimes asks about it:

• •If b² − 4ac > 0: two real solutions
• •If b² − 4ac = 0: one real solution (repeated root)
• •If b² − 4ac < 0: no real solutions

Factoring Special Forms

Memorize these — they show up repeatedly:

• •Difference of squares: a² − b² = (a + b)(a − b)
• •Perfect square trinomial: a² + 2ab + b² = (a + b)²
• •Perfect square trinomial: a² − 2ab + b² = (a − b)²

Example: x² − 25 = 0

Recognize it as a difference of squares: (x + 5)(x − 5) = 0, so x = 5 or x = −5.

The ACT uses these patterns to create questions that look complicated but are actually fast — if you recognize the form.

Non-Polynomial Equations

Not every equation on the ACT involves polynomials. You'll also see equations with radicals, absolute values, exponentials, and rational expressions. The universal strategy: convert them back to polynomial form so you can solve with the tools you already know.

Radical Equations

When the variable is under a square root, isolate the radical and square both sides:

Example: √(2x + 3) = 5

1. 1.Square both sides: 2x + 3 = 25
2. 2.Solve: 2x = 22, so x = 11

Critical step: Always check your answer by plugging it back into the original equation. Squaring both sides can introduce extraneous solutions — answers that satisfy the squared equation but not the original.

Absolute Value Equations

An absolute value equation splits into two cases:

Example: |x − 4| = 7

• •Case 1: x − 4 = 7 → x = 11
• •Case 2: x − 4 = −7 → x = −3

Both solutions are valid (check by substituting). The ACT will sometimes try to trick you by giving only one of the two solutions as an answer choice — make sure you consider both cases.

Rational Equations

Equations with the variable in the denominator require you to multiply through by the denominator to clear fractions:

Example: 3/(x − 1) = 6

1. 1.Multiply both sides by (x − 1): 3 = 6(x − 1)
2. 2.Distribute: 3 = 6x − 6
3. 3.Solve: 9 = 6x, so x = 3/2

Always check that your answer doesn't make a denominator zero. If it does, it's an extraneous solution.

Exponential Equations

When the variable is in the exponent, try to express both sides with the same base:

Example: 2^(x+1) = 8

1. 1.Rewrite 8 as 2³: 2^(x+1) = 2³
2. 2.Since the bases are equal, the exponents must be equal: x + 1 = 3
3. 3.Solve: x = 2

If you can't match bases, the ACT usually provides answer choices you can test — plug them in and see which one works.

Systems of Equations

A system of equations is two or more equations with two or more unknowns. The ACT tests these regularly, and you need to be comfortable with two methods.

Substitution

Best when one equation already has a variable isolated:

System:

• •y = 2x + 1
• •3x + y = 11

Since y is already isolated in the first equation, substitute into the second:

• •3x + (2x + 1) = 11
• •5x + 1 = 11
• •5x = 10
• •x = 2, then y = 2(2) + 1 = 5

Elimination

Best when both equations are in standard form (ax + by = c):

System:

• •2x + 3y = 12
• •4x − 3y = 6

The y terms are already set up to cancel — add the equations:

• •6x = 18
• •x = 3, then substitute back: 2(3) + 3y = 12, so y = 2

When the coefficients don't naturally cancel, multiply one or both equations by a constant first:

System:

• •3x + 2y = 16
• •5x + 4y = 28

Multiply the first equation by −2: −6x − 4y = −32

Add to the second: −x = −4, so x = 4, then y = 2

Which Method to Use?

• •If one variable is already isolated → substitution
• •If the equations are in standard form → elimination
• •If you're stuck → try plugging in answer choices

The ACT provides answer choices as ordered pairs (x, y). Plugging them into both equations to check is always a valid strategy, especially when the algebra feels overwhelming.

The #1 ACT Algebra Strategy: Identify the Equation Type First

Here's the single most important piece of advice for ACT Math algebra: before you start solving, take 5 seconds to identify what type of equation you're looking at.

• •Is it linear? → Isolate the variable.
• •Is it a quadratic? → Factor or use the quadratic formula.
• •Does it have a radical? → Isolate and square.
• •Is the variable in the denominator? → Multiply to clear fractions.
• •Is it a system? → Choose substitution or elimination.

Students who rush into solving without identifying the equation type make more errors and waste more time. Students who pause, classify, and then select their approach work faster and more accurately.

When All Else Fails: Plug In the Answers

The ACT is a multiple-choice test. This is your superpower. If you don't know how to solve an equation algebraically, you can test each answer choice by substituting it into the equation and checking if it works.

This isn't cheating — it's strategic. Start with the middle answer choice (typically choice C). If it's too big, try a smaller option. If it's too small, try a larger one. This process takes 30–60 seconds and works on every equation-solving question on the test.

Remember: Most Hard Problems Combine Basic Skills

The hardest ACT Math problems rarely introduce genuinely new concepts. Instead, they combine two or three basic skills in a single question — for example, setting up a system of equations from a word problem, or factoring a quadratic that's hidden inside a rational expression.

If a problem looks intimidating, break it down. Identify the skills involved. Handle them one step at a time. The building blocks are the same ones covered in this guide.

Practice Makes Permanent

Understanding algebra concepts isn't enough — you need to practice them under timed conditions until the approaches become automatic. On test day, you don't have time to think about how to solve 3x + 7 = 22. That should be muscle memory.

Our ScoreJump course covers all of these algebra topics in depth with video instruction and guided practice. If you need structured, section-by-section ACT prep, our ACT Blueprint program builds a personalized plan around your student's specific math skill gaps.

Quick Reference: ACT Algebra Cheat Sheet

Master these patterns and you'll be ready for the algebra questions the ACT throws at you — no matter how they're disguised.