ACT Math Overview

Welcome back to another Fiveable ACT guide! This guide will be covering ACT Algebra, which will make up 12-15% of your math exam, or about 7-9 of 60 questions. You should be able to manipulate and analyze basic algebraic equations. Let’s break down the skills you need to be successful in this topic area and go over some practice questions. Stay focused and determined! 🎯

🗺️ Main Algebra Topic Areas

ACT has provided a list of skills that may be tested under the Algebra subcategory. We’ve organized them into two main topic areas to ensure you cover everything you need to know. 

🧮 Basic Expressions and Equations

• Creating representations and solving multiple types of expressions is key for this topic.

🔢 Higher Level Algebraic Functions

• These types of questions will test your ability to analyze and manipulate algebraic functions. 

• The ACT will test your ability to work with many types of functions, including linear, polynomials (like quadratic and cubic equations), exponential, logarithmic, and radical equations. 

🧮  Basic Expressions and Equations

This area builds on the basic algebra you learned in school. Let's cover some tips and tricks for tackling these questions!

🧠 What You Need to Know:  Basic Expressions and Equations

For the first group of skills, 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 with variable isolation. 

• Identify characteristics of a function, such as slope and intercepts. 

  • Slope is the rate of change of a function, and can usually be calculated as (change in y ) / (change in x). You may have heard of it being referred to rise/run!

  • The Y-intercept is where the function crosses or touches the y-axis.

  • The X-intercept is where the function crosses or touches the x-axis.

🤓 Applying Your Knowledge:  Basic Expressions and Equations

Simplifying Expressions Practice

Simplifying expressions is a critical skill for algebra and is usually done with the main mathematical operations. Sometimes, if you don’t see your answer as a choice, simplification can reveal the truth! 🔮

Image Courtesy of ACT’s 2023 ACT Test Guide

The correct answer is J. 

First, we need to combine like terms. Since -4x3 and -12x3 have the same term (x3), we can add those together: -4x3 + -12x3 = -16x3 . Because we are adding the terms, make sure to only add the coefficients, and not to change the variable and its exponent. We now have the expression -16x3 + 9x2!

Solving Equations Practice

The study of algebra is based on being able to solve an equation. Let’s use this example to hone our skills! 📝

Image Courtesy of ACT’s 2023 ACT Test Guide

The correct answer is K. 

When you work through solving an equation, you should write down your work. It helps keep track of our steps and allows us to quickly identify our mistakes. Check out the table below, which explains the process of isolating the variable and solving for our x value!

Manipulating Expressions Practice

Manipulating expressions can quickly become complicated if your work is not organized. You only get better with practice, so let’s start! 🏁

Image Courtesy of ACT’s 2023 ACT Test Guide

The correct answer is E. 

Don’t panic when you see x ≠ ± y! This just means that the variable x cannot equal the variable y, positive or negative. This just protects the equation from having an undefined fraction (#/0) so we can work through the problem without worrying. 😌 

To add the two fractions together, we have to make sure they have the same denominator. In this case, the simplest common denominator can be found by multiplying the two denominators together: (x+y)(x-y). Follow the steps in the table below to get our final answer!

🔢 Higher Level Algebraic Functions

Let's take it to the next level by incorporating the skills you learned above other types of equations. 🚀

🧠 What You Need to Know:  Higher Level Algebraic Functions

For the second group of skills, you should be able to:

• Manipulate polynomial equations.

• Solve equations involving squares, cubes, square roots, and cube roots.

• Solve and identify inequalities.

  • These usually include the following words and symbols: < (less than) , > (greater than) , ≤ (less than or equal to) , ≥ (greater than or equal to)

  • If the equation is < or > its conditions, it will be marked by a dashed line because the solution to the equation will not be included in the conditions. - - - - - >

  • If the equation is ≤ or ≥ its conditions, it will be marked by a solid line, indicating that a solution to the equation is included in the conditions. --->

• Solve systems of equations.

  • A system of equations consists of more than one equation. The solution to the system is the point or area that meets the conditions of both equations.

  • This is usually done with substitution or elimination, or by observing a graph to find where the two equations share a solution.

🤓 Applying Your Knowledge:  Higher Level Algebraic Functions

Quadratic Equations

Quadratic equations have a variable, usually x, that is squared. Therefore, there can be 0, 1, or 2 solutions to a single equation. Let’s practice! 

Image Courtesy of ACT’s 2023 ACT Test Guide

The correct answer is A.  

To solve a quadratic equation in the y = ax^2 + bx + c format, you can either factor or use the quadratic formula: 

In this case, we can factor, which would also be the easier method to use. There are many ways to factor, so use whichever is most comfortable for you! Follow along below.  

But we’re not done yet! Now we have to choose the solution that is greater than the other. Since ⅚ > ¾, the answer is ⅚. ⭐

System of Inequalities Practice

This question will combine your inequality identification skills as well as your system of equations skills! You got this! 🍀

Image Courtesy of ACT’s 2023 ACT Test Guide

The correct answer is K. 

Let's look at the first equation, and put it in slope intercept form. By doing so, we get y ≤ –x/2 + 3. This is a line! We now know that the slope of the graph (m) is -½, and the y-intercept is (0,3). Let's sketch this out with a solid line and shade underneath, since the equation is less than or equal to. 

We can determine from the answer choices that the second equation must be a circle, so let's get the variables on one side and simplify the equation. We get x2 + y2 > 4. Since the base equation for a circle is x2 + y2 = r2, we can now sketch out a circle of radius 2 at the origin. Now let’s sketch a circle has a dotted border and shade outside of the circle, since the circle’s radius must be greater than 2. 

Your sketch should look something like this! Since the area inside the circle and the area above the line do not meet the conditions for both equations in the system, the part that is double shaded above should match the correct answer choice, K. Amazing work! 😁

🌟 Closing

Congratulations! You’ve finished the ACT Math Algebra prep 🙌 You should have a better understanding of the Math sections for the ACT, topic highlights, what you will have to be able to do in order to succeed, as well as have seen some practice questions that put the concepts in action. Good luck studying for the ACT Math section!! 👏

Need more ACT resources? Check out our other ACT Math Guides with practice problems. Keep up the good work 🥳!!