ACT Math: Preparing for Higher Math: Algebra

ACT Math Overview

This guide covers ACT Algebra, which makes up 12-15% of the math exam (about 7-9 out of 60 questions). The questions test your ability to manipulate and analyze algebraic equations, from combining like terms all the way up to systems of inequalities. Below, you'll find the key skills and worked practice problems to help you prepare.

🗺️ Main Algebra Topic Areas

ACT lists several skills under the Algebra subcategory. They fall into two main areas:

🧮 Basic Expressions and Equations

• Creating, simplifying, and solving expressions and equations using core algebra skills.

🔢 Higher Level Algebraic Functions

• Analyzing and manipulating more complex algebraic functions, including linear, polynomial (quadratic, cubic), exponential, logarithmic, and radical equations.

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🧮 Basic Expressions and Equations

This area builds on the algebra fundamentals you've already learned. Here are the core skills and some practice to sharpen them.

🧠 What You Need to Know: Basic Expressions and Equations

You should be able to:

• Set up a basic expression and simplify it 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 is the rate of change of a function, calculated as (change in y) / (change in x). You might know this as "rise over run."
  • The y-intercept is the point where the function crosses the y-axis (where $x = 0$).
  • The x-intercept is the point where the function crosses the x-axis (where $y = 0$).

🤓 Applying Your Knowledge: Basic Expressions and Equations

Simplifying Expressions Practice

Simplifying expressions is a foundational algebra skill. On the ACT, if your answer doesn't match any of the choices, try simplifying further.

Image Courtesy of ACT's 2023 ACT Test Guide

The correct answer is J.

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

When combining like terms, only add or subtract the coefficients. Don't change the variable or its exponent.

Solving Equations Practice

Solving equations by isolating the variable is one of the most tested algebra skills on the ACT.

Image Courtesy of ACT's 2023 ACT Test Guide

The correct answer is K.

Write down every step when solving equations. It keeps you organized and makes it easy to spot mistakes.

Manipulating Expressions Practice

Manipulating expressions with fractions can get messy fast, so keep your work organized step by step.

Image Courtesy of ACT's 2023 ACT Test Guide

The correct answer is E.

The condition $x \neq \pm y$ just means $x$ can't equal $y$ or $-y$. This prevents a denominator of zero, so you don't need to worry about it while solving.

To add two fractions, they need a common denominator. Here, multiply the two denominators together to get $(x+y)(x-y)$. Then adjust each fraction's numerator accordingly and combine.

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🔢 Higher Level Algebraic Functions

This section builds on the basics above and applies them to more complex equation types.

🧠 What You Need to Know: Higher Level Algebraic Functions

You should be able to:

• Manipulate polynomial equations (factoring, expanding, simplifying).
• Solve equations involving squares, cubes, square roots, and cube roots.
• Solve and identify inequalities.
  • Key symbols: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to).
  • If the inequality uses < or >, graph it with a dashed line because points on the line are not included in the solution.
  • If the inequality uses ≤ or ≥, graph it with a solid line because points on the line are included.
• Solve systems of equations.
  • A system has more than one equation. The solution is the point (or region) that satisfies all equations simultaneously.
  • Common methods: substitution, elimination, or reading the intersection from a graph.

🤓 Applying Your Knowledge: Higher Level Algebraic Functions

Quadratic Equations

Quadratic equations have a squared variable (usually $x^2$), which means they can have 0, 1, or 2 solutions.

Image Courtesy of ACT's 2023 ACT Test Guide

The correct answer is A.

To solve a quadratic in the form $ax^2 + bx + c = 0$, you can either factor or use the quadratic formula:

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

For this problem, factoring is the quicker approach:

Common mistake to watch for: After factoring, don't forget to check the signs. The factors $(4x-3)(6x+5) = 0$ give $x = \frac{3}{4}$ and $x = -\frac{5}{6}$. Double-check by plugging your answers back into the original equation if you have time.

System of Inequalities Practice

This question combines inequality graphing with systems of equations.

Image Courtesy of ACT's 2023 ACT Test Guide

The correct answer is K.

Here's how to work through it:

1. Rearrange the first inequality into slope-intercept form. You get $y \leq -\frac{x}{2} + 3$. The slope is $-\frac{1}{2}$ and the y-intercept is $(0, 3)$. Since the symbol is ≤, draw a solid line and shade below it.

2. Rearrange the second inequality. Move the variables to one side to get $x^2 + y^2 > 4$. The standard form of a circle is $x^2 + y^2 = r^2$, so this is a circle centered at the origin with radius $r = 2$. Since the symbol is >, draw a dashed circle and shade outside it (the solution includes points whose distance from the origin is greater than 2).

3. Find the overlap. The solution to the system is the region that satisfies both inequalities: below the solid line and outside the dashed circle. That double-shaded region matches answer choice K.

Your sketch should look something like this. The double-shaded region is the answer.

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🌟 Closing

You've covered the key algebra skills tested on the ACT Math section. To recap, make sure you're comfortable with:

• Simplifying and combining like terms
• Solving linear equations by isolating the variable
• Adding/subtracting algebraic fractions
• Factoring quadratics and using the quadratic formula
• Graphing and solving systems of inequalities

Need more practice? Check out the other ACT Math Guides for additional problems and strategies.