ACT Math: Preparing for Higher Math – Algebra

ACT Math Overview

Welcome to this comprehensive guide on the Algebra section of the ACT Math test. Algebra plays a crucial role in the ACT, contributing 12-15% of the total math exam, which translates to about 7-9 questions out of 60. Mastering algebraic concepts is essential for success in this section. This guide will break down the key algebra skills you need, provide strategies, and offer practice questions to help you excel. Let’s dive in and tackle the algebra section of the ACT with confidence.

Reminder: Don’t forget your calculator! You are allowed to use a permitted calculator throughout the entire ACT Math section.

Main Algebra Topic Areas

The ACT has categorized algebra into several key skills, which can be grouped into two main topic areas:

1. Basic Expressions and Equations
2. Higher-Level Algebraic Functions

These categories encompass a variety of algebraic concepts, from simplifying expressions to solving complex equations. Below, we’ll explore each area in detail, along with strategies and practice questions.

Basic Expressions and Equations

This section covers foundational algebra skills that you’ve likely encountered in high school. These skills are essential for solving basic algebra problems on the ACT.

What You Need to Know: Basic Expressions and Equations

For this portion of the exam, you should be proficient in the following:

• Setting Up and Simplifying Expressions: You should be able to create algebraic expressions and simplify them by combining like terms.
• Graphing and Modeling Expressions: Understanding how to graph equations and model real-world situations using algebraic expressions is key.
• Substituting Values: You’ll need to substitute specific values into equations and expressions to find solutions.
• Solving Basic Equations: Be comfortable with isolating variables to solve equations.
• Identifying Characteristics of Functions: Understand concepts such as slope, y-intercept, and x-intercept:
  • Slope: This represents the rate of change of a function and is typically calculated as the change in y divided by the change in x.
  • y-intercept: The point where the function crosses the y-axis.
  • x-intercept: The point where the function crosses the x-axis.

Applying Your Knowledge: Basic Expressions and Equations

Simplifying Expressions Practice

Simplifying expressions is a fundamental skill in algebra and involves combining like terms and using basic arithmetic operations.

Example:

Simplify the expression:

$7x^2 – 4x^2 + 3x – 2x + 5$

Solution:

First, combine the like terms:

• Combine the
  $x^2$ terms: $7x^2 – 4x^2 = 3x^2$
  
• Combine the
  $x$ terms: $3x – 2x = 1x$
  

The simplified expression is:

$3x^2+x+5$

Other examples$3x^2 + x + 5$

Image from act.org 

The correct answer is A. 

Solving Equations Practice

Being able to solve algebraic equations is crucial for the ACT Math section. Let’s go through an example.

Example:

Solve for $\mathrm{x:}$

$$$2(3x – 4) = 5(x + 2)$

Solution:

1. Distribute the numbers outside the parentheses:

   $6x – 8 = 5x + 10$

    

2. Get all the x-terms on one side:

   $6x – 5x = 10 + 8$
   
   
   $x = 18$

    

The solution is

$x=18.$

$\mathrm{<span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;">Other\; examples:</span>}$

Image from act.org

The correct answers are:
9. E
10. C

Manipulating Expressions Practice

Manipulating algebraic expressions often involves working with fractions, exponents, and factoring.

Example:

Simplify the expression:

$\frac{2x}{x^2 – 9} + \frac{3x}{x^2 – 9}$

​

Solution:

Since both fractions have the same denominator, you can combine them:

$\frac{2x + 3x}{x^2 – 9} = \frac{5x}{x^2 – 9}$

​This is the simplified expression.

Some more examples:

Image from act.org

The correct answers are:
13. B
14. E

Higher-Level Algebraic Functions

In this section, we move beyond the basics and explore more complex algebraic functions. These include polynomial equations, inequalities, and systems of equations.

What You Need to Know: Higher-Level Algebraic Functions

For this portion of the ACT Math section, you should be familiar with the following:

• Manipulating Polynomial Equations: This includes working with quadratic, cubic, and other higher-order polynomials.

Image from act.org

• Simplifying and Solving Equations Involving Roots: Be prepared to solve equations involving square roots, cube roots, and their corresponding powers.

Here’s an example:

Image from act.org

The correct answer is:
12. C

• Solving and Identifying Inequalities: You should understand how to solve inequalities and graph them on a number line or coordinate plane.
  • < and >: Represent less than and greater than, respectively. These are shown with dashed lines when graphed because the solution does not include the boundary.
  • ≤ and ≥: Represent less than or equal to, and greater than or equal to, respectively. These are shown with solid lines because the solution includes the boundary.

Example: 

Image from act.org

The correct answer is:
8. E

• Solving Systems of Equations: This involves finding the solution to multiple equations simultaneously, often by substitution, elimination, or graphing.

Example:

The correct answer is:
32. D

Image from act.org

Applying Your Knowledge: Higher-Level Algebraic Functions

Quadratic Equations

Quadratic equations are a common feature in the ACT Math section. These equations involve a variable squared and can have up to two solutions.

Example:

Solve the quadratic equation:

$x^2 – 5x + 6 = 0$

Solution:

This equation can be factored:

$(x – 2)(x – 3) = 0$

Set each factor equal to zero:

$x – 2 = 0 \quad \text{or} \quad x – 3 = 0$

 $x = 2 \quad \text{or} \quad x = 3$

The solutions are

$x = 2$

 and

$x = 3$

.

System of Inequalities Practice

System of inequalities questions require you to solve for conditions that satisfy multiple inequalities simultaneously.

Example:

Solve the system of inequalities:

$$\begin{aligned} &2x + y \leq 4 \\ &x – y > 2 \end{aligned}$$

​Solution:

First, graph each inequality on a coordinate plane:

1. For

   $2x + y \leq 4$

    

   • Rearrange to slope-intercept form:
     $y \leq -2x + 4$

      

   • This is a line with a slope of -2 and y-intercept at 4, shaded below because of the inequality.

2. For

   $x – y > 2$

    

   • Rearrange to slope-intercept form:
     $y < x – 2$

      

   • This is a line with a slope of 1 and y-intercept at -2, shaded below the line.

Image from desmos.com

The solution to the system of inequalities is the region where the shaded areas overlap.

Conclusion

Congratulations on completing this guide to the Algebra section of the ACT Math exam. You’ve now reviewed the key algebraic concepts that will appear on the test, from basic expressions to more advanced functions. By practicing these skills and applying the strategies outlined in this guide, you’ll be well-prepared to tackle the algebra questions on the ACT.

Remember, consistent practice is the key to mastering these concepts. Keep working through practice problems, take timed practice tests, and review any mistakes you make along the way. With dedication and preparation, you can achieve your target score on the ACT Math section.

Good luck, and keep up the hard work as you prepare for the ACT!