ACT Math: Preparing for Higher Math: Algebra

TL;DR

ACT Math is 45 questions (41 scored) in 50 minutes, and calculators are permitted throughout. Algebra falls under the "Preparing for Higher Math" reporting category and accounts for roughly 12–15% of the scored questions. You need to be comfortable with simplifying expressions, solving equations, working with quadratics, and handling systems of equations and inequalities.

ACT Math: Algebra Overview

This guide covers the Algebra subcategory of ACT Math's "Preparing for Higher Math" domain. Questions range from combining like terms to solving systems of inequalities. The two main skill areas are:

• Basic Expressions and Equations — creating, simplifying, and solving using core algebra skills
• Higher Level Algebraic Functions — working with linear, polynomial, exponential, logarithmic, and radical equations

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Basic Expressions and Equations

What You Need to Know

You should be able to:

• Set up and simplify expressions by combining like terms
• Graph and model expressions
• Substitute values into an equation to find a solution
• Solve basic equations by isolating the variable
• Identify characteristics of a function, such as slope and intercepts
  • Slope = (change in y) ÷ (change in x), also called "rise over run"
  • y-intercept: where the function crosses the y-axis (x = 0)
  • x-intercept: where the function crosses the x-axis (y = 0)

Simplifying Expressions

Simplifying expressions is a foundational skill. If your answer doesn't match any choice, try simplifying further.

Example: Simplify $-4x^3 + 9x^2 - 12x^3$

1. Identify like terms: $-4x^3$ and $-12x^3$ both have the variable $x^3$.
2. Add the coefficients: $-4 + (-12) = -16$. The variable and exponent stay the same.
3. Simplified result: $-16x^3 + 9x^2$

Solving Equations

Isolating the variable is one of the most tested algebra skills. Write every step to stay organized.

Example: Solve $-3(4x - 5) = 2(1 - 5x)$

Manipulating Expressions with Fractions

To add two algebraic fractions, find a common denominator first.

Example: Add $\frac{1}{x+y} + \frac{1}{x-y}$ (where $x \neq \pm y$)

• Common denominator: $(x+y)(x-y)$
• Rewrite each fraction with that denominator, then combine numerators
• The condition $x \neq \pm y$ simply prevents a zero denominator — you don't need to do anything extra with it while solving

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Higher Level Algebraic Functions

What You Need to Know

You should be able to:

• Manipulate polynomial equations (factoring, expanding, simplifying)
• Solve equations involving squares, cubes, square roots, and cube roots
• Solve and graph inequalities
  • < or > → dashed boundary line (points on the line are not included)
  • ≤ or ≥ → solid boundary line (points on the line are included)
• Solve systems of equations using substitution, elimination, or graphing

Quadratic Equations

Quadratic equations have a squared variable ($x^2$) and can have 0, 1, or 2 solutions.

To solve $ax^2 + bx + c = 0$, factor when possible or use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Example: Find the larger solution of $24x^2 + 2x = 15$

Since $\frac{3}{4} > -\frac{5}{6}$, the larger solution is $\frac{3}{4}$.

System of Inequalities

These questions combine inequality graphing with systems of equations.

Example: Find the region satisfying both $y \leq -\frac{x}{2} + 3$ and $x^2 + y^2 > 4$

1. First inequality: Slope $= -\frac{1}{2}$, y-intercept $= (0, 3)$. Use a solid line (≤) and shade below it.
2. Second inequality: This is a circle centered at the origin with radius 2. Use a dashed circle (>) and shade outside it (points farther than 2 units from the origin).
3. Overlap: The solution is the region that satisfies both — below the solid line and outside the dashed circle.

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Key Algebra Skills: Quick Reference

Study Tips

• Practice factoring quadratics quickly — it saves time compared to always using the quadratic formula.
• For systems of inequalities, sketch the graph even on scratch paper. Visualizing the shaded regions reduces errors.
• Write out every algebraic step. One sign error can cost you the question.
• Use your calculator to check arithmetic, but set up the algebra by hand first.