Inequalities & Absolute Value

Historical Context & Motivation

Mathematics has always needed ways to express relationships that go beyond simple equality. When ancient merchants wanted to say that a shipment must weigh at least a certain amount, or when engineers needed to specify that a measurement must fall within a certain tolerance, they needed a language richer than the equals sign alone. Inequalities and absolute value developed over centuries to fill that need, eventually becoming essential tools in algebra, calculus, and standardized tests like the ACT.

On the ACT, inequalities and absolute value questions test whether you can think about ranges of solutions rather than single answers. The central question these concepts address is: What set of values satisfies a given condition? Understanding this question—and the techniques to answer it—will help you tackle a significant portion of the ACT Math section with confidence.

Core Principles & Definitions

Before you can solve inequality and absolute value problems quickly, you need a solid grasp of the foundational ideas that govern how they work. These principles are the building blocks for every problem you will see on the ACT.

Visual Explanation — Number Line Representations

One of the most powerful ways to understand inequalities and absolute value is to visualize them on a number line. The diagram below shows how different inequality types map to shaded regions. Open circles (○) indicate that the endpoint is not included (strict inequalities: < or >), while closed circles (●) indicate the endpoint is included (non-strict: ≤ or ≥).

The key visual pattern to remember is this: absolute value inequalities with less than (< or ≤) produce a single connected segment on the number line, because the values are "trapped" between two bounds. Absolute value inequalities with greater than (> or ≥) produce two separate rays pointing outward, because the values must be far enough from the center. On the ACT, if you can quickly sketch a number line in the margin, you can often identify the correct answer choice by matching it to the right shape.

Mathematical Framework

Now let's formalize the rules you need. These equations and properties are the tools you will apply on test day. Make sure you understand not just what each rule says but why it works—that way you won't need to rely on pure memorization.

Detailed Breakdown — Types of Problems on the ACT

On the ACT, inequality and absolute value questions appear in several distinct forms. The diagram below organizes these problem types into a decision tree that will help you quickly identify which technique to apply when you see a question on test day.

Worked Example

Let's walk through an ACT-style problem from start to finish. Pay close attention to where the sign flip occurs and how the absolute value splits into two cases.

Common Mistakes & How to Avoid Them

Even students who understand the rules still lose points on the ACT because of a handful of predictable mistakes. Learning what these traps look like—and how to avoid them—is just as important as learning the rules themselves.

Connection to Advanced Math

The inequality and absolute value skills you build for the ACT don't stop being useful once you turn in your answer sheet. These concepts are stepping stones to more powerful mathematical ideas you'll encounter in college-level courses. Understanding these connections now can give you a preview of what lies ahead.

For now, focus on mastering the core techniques. The beauty of these concepts is that the same fundamental logic—comparing quantities, handling distance, and reasoning about ranges—scales all the way from a high school math test to graduate-level analysis.

Practice Problems

Work through these five problems in order. They increase in difficulty, mirroring the range you will see on the actual ACT. Try each one on your own before reading the answer.

Lesson Summary