Absolute Value Inequalities

What Are Absolute Value Inequalities?

Absolute value inequalities are mathematical statements that combine the concept of absolute value with an inequality symbol like less than (">") or greater than ( "<"). While an absolute value equation like $|x|=5$ asks, "Which numbers are exactly 5 units from zero?" (the answers are $5$ and $-5$), an inequality like $|x|<5$ asks, "Which numbers are less than 5 units from zero?" This question has a whole range of answers.

Remember, the absolute value of a number, written as $|a|$, represents its distance from $0$ on the number line. Since distance is always positive or zero, the absolute value of any number is always non-negative. For example, $|7|=7$ and $|-7|=7$.

When we mix this idea of distance with inequalities—$<$, $>$, $\le$, and $\ge$—we are no longer looking for one or two specific points. Instead, we are looking for entire sets of numbers that satisfy a condition. The solution to an absolute value inequality is typically a range of values, which we can show on a number line or write using interval notation.

The Two Core Types: 'Less Than' and 'Greater Than'

Almost every absolute value inequality problem you'll encounter can be boiled down to one of two fundamental types. Understanding the difference is the key to solving them correctly. The secret is to think about what the inequality means in terms of distance.

'Less Than' is an 'AND' Statement

Consider the inequality $|x|<4$. This asks for all numbers whose distance from zero is less than 4. If you picture a number line, these are all the numbers squished between $-4$ and $4$. For a number $x$ to be in this group, it must satisfy two conditions at once: it must be greater than $-4$ AND it must be less than $4$. This leads to a compound inequality.

This rule also applies to $\le$. The inequality $|A|\le c$ becomes $-c\le A\le c$. Think of these as "sandwich" inequalities, where the expression $A$ is trapped between two values.

'Greater Than' is an 'OR' Statement

Now consider $|x|>4$. This asks for all numbers whose distance from zero is greater than 4. On a number line, these numbers are not in the middle. They are in two separate regions: all the numbers to the right of $4$ OR all the numbers to the left of $-4$. This cannot be written as a single sandwich. It requires two separate statements connected by "OR".

This rule also applies to $\ge$. The inequality $|A|\ge c$ becomes $A\ge c$ OR $A\le -c$. Think of these as "breakaway" inequalities that describe two distinct sets of numbers.

How to Solve Absolute Value Inequalities: A Step-by-Step Guide

Solving these inequalities follows a predictable pattern. By turning the problem into a standard linear inequality (or two), you can use the algebra skills you already have. Follow these steps to find the solution every time.

1. Isolate the Absolute Value Expression. Before you can apply any rules, you must get the absolute value part, $|...|$, completely by itself on one side of the inequality. Use addition, subtraction, multiplication, or division to move all other terms to the opposite side.
2. Check the Constant. Look at the number on the other side of the inequality. If it's positive, proceed to the next step. If it's zero or negative, you have a special case (covered in the next section).
3. Split into Two Cases. Based on the inequality symbol, rewrite the problem without absolute value bars.
   • For $<$ or $\le$ (Less Th-AND), write a single compound inequality: $-c<A<c$.
   • For $>$ or $\ge$ (Great-OR), write two separate inequalities: $A>c$ OR $A<-c$. Don't forget to flip the inequality sign for the negative case!
4. Solve the Inequality (or Inequalities). Solve the resulting linear inequalities to find the possible values for your variable. If you have a compound inequality, remember to perform the same operation on all three parts. If you have two separate inequalities, solve them independently.
5. Graph the Solution on a Number Line. A visual representation is extremely helpful. Use open circles ($\circ$) for $<$ and $>$ to show the endpoint is not included. Use closed circles ($\bullet$) for $\le$ and $\ge$ to show the endpoint is included.
6. Write the Solution in Interval Notation. This is the standard way to express your final answer. Use parentheses $()$ for open circles and brackets $[])$ for closed circles. For "OR" problems with two separate regions, use the union symbol ($\cup$) to connect the two intervals.

Worked Examples: Putting It All Together

Let's walk through a few examples to see the step-by-step process in action. We will apply the rules for isolating, splitting, solving, and graphing.

What About Special Cases? Zero and Negative Numbers

What happens if, after isolating the absolute value, the number on the other side isn't positive? This is where you can save a lot of work by stopping and thinking about the definition of absolute value.

Case 1: Comparing to a Negative Number

Remember, absolute value represents distance, which can never be negative. The value of $|A|$ is always $0$ or positive.

• $|A|>-2$ or $|A|\ge -2$: Since $|A|$ is always non-negative, it is always greater than any negative number. The solution is all real numbers, or $(-\mathrm{\infty },\mathrm{\infty })$.
• $|A|<-2$ or $|A|\le -2$: Since $|A|$ can never be negative, it can never be less than a negative number. There is no solution.

For example, if you have $|4x-9|<-3$, you can immediately say there is no solution without doing any more algebra.

Case 2: Comparing to Zero

Comparing to zero also follows a special logic.

• $|A|>0$: The absolute value is greater than zero for everything except when the expression inside is zero. The solution is everything except where $A=0$.
• $|A|\ge 0$: The absolute value is always greater than or equal to zero. The solution is all real numbers.
• $|A|<0$: The absolute value can never be negative, so it can never be less than zero. There is no solution.
• $|A|\le 0$: The only way this can be true is if the absolute value is exactly zero. The solution is found by solving $A=0$. For example, the only solution to $|x+3|\le 0$ is $x=-3$.

Common Mistakes to Avoid

Absolute value inequalities can be tricky, and there are a few common pitfalls that students often fall into. Be on the lookout for these errors in your own work.

• Forgetting to Isolate First: A very common mistake is to split the inequality before the absolute value expression is alone. For example, in $2|x+1|<8$, you must divide by $2$ first to get $|x+1|<4$ before you split it into $-4<x+1<4$.
• Mixing Up the AND/OR Rules: Students sometimes apply the "sandwich" rule to a "greater than" problem or vice versa. Remember the mnemonic: Less Th-AND (a sandwich) and Great-OR (two separate pieces).
• Incorrectly Negating in the 'OR' Case: When you split $|A|>c$ into $A>c$ OR $A<-c$, it is crucial to both flip the inequality sign AND make the constant negative in the second piece. Forgetting to do one of these will lead to a wrong answer.
• Distributing into Absolute Value Bars: You cannot distribute a number into absolute value bars like you do with parentheses. The expression $3|x-2|$ is not equal to $|3x-6|$. You must isolate the absolute value first.
• Errors in Interval Notation: A small mix-up between a parenthesis $()$ and a bracket $[])$ can make a correct answer incorrect. Remember: brackets for $\le$ and $\ge$ (endpoints included), and parentheses for $<$ and $>$ (endpoints excluded). Infinity $(\mathrm{\infty })$ always gets a parenthesis.

Quick Summary and Reference

This table summarizes the main rules for solving absolute value inequalities after the absolute value expression has been isolated and where $c$ is a positive number.