Absolute Value Inequalities
Absolute value inequalities are inequalities in algebra that involve algebraic expressions with absolute value symbols and inequality symbols. In simple words, we can say that an absolute value inequality is an inequality with an absolute value symbol in it. It can be solved using two methods of either the number line or the formulas. An absolute value inequality is a simple linear expression in one variable and has symbols such as >, <, >, <.
In this article, we will learn the concept of absolute value inequalities and the methods to solve them. We will mainly focus on the linear absolute value inequalities and discuss how to graph them with the help of various solved examples for a better understanding of the concept.
1.  What Are Absolute Value Inequalities? 
2.  Solving Absolute Value Inequalities 
3.  Graphing Absolute Value Inequalities 
4.  Absolute Value Inequalities Formulas 
5.  FAQs on Absolute Value Inequalities 
What Are Absolute Value Inequalities?
An absolute value inequality is an inequality that involves an absolute value algebraic expression with variables. Absolute value inequalities are algebraic expressions with absolute value functions and inequality symbols. That is, an absolute value inequality can be one of the following forms (or) can be converted to one of the following forms:
 ax + b < c
 ax + b > c
 ax + b < c
 ax + b > c
So the absolute value inequalities are of two types. They are either of lesser than or equal to or are of greater than or equal to forms. The two varieties of inequalities are as follows.
 one with < or ≤
 one with > or ≥
Solving Absolute Value Inequalities
In this section, we will learn to solve the absolute value inequalities. Here is the procedure for solving absolute value inequalities using the number line. The procedure to solve the absolute value inequality is shown stepbystep along with an example for a better understanding.
Example: Solve the absolute value inequality x+2 < 4
Solution:
Step 1: Assume the inequality as an equation and solve it.
Convert the inequality sign "<" in our inequality to "=" and solve it.
⇒ x + 2 = 4
Removing the absolute value sign on the left side, we get + sign on the other side.
⇒ x + 2 = + 4
This results in two equations, one with "+" and the other with "".
⇒ x+2 = 4 and x+2 = 4
⇒ x = 2 and x = 6
Step 2: Represent the solutions from Step 1 on a number line in order.
Here, we can see that the number line is divided into 3 parts/intervals.
Step 3: Take a random number from each of these intervals and substitute it with the given inequality. Identify which of these numbers actually satisfies the given inequality.
Interval 
Random Number  Checking the given inequality with a random number 

(∞, 6)  7  7+2 < 4 ⇒ 5 < 4 ⇒ This is False 
(6, 2)  0  0+2 < 4 ⇒ 2 < 4 ⇒ This is True 
(2, ∞)  3  3+2 < 4 ⇒ 5 < 4 ⇒ This is False 
Step 4: The solution of the given inequality is the interval(s) which leads to True in the above table
Therefore, the solution of the given inequality is, (6, 2) or (6 < x < 2). This procedure is summarized in the following flowchart.
Note:
 If the problem was x+2 < 4, then the solution would have been [6, 2] (or) 6 < x < 2. i.e.,
 If x + 2 < 4 ⇒ 6 < x < 2
 If x+2 ≤ 4 ⇒ 6 ≤ x ≤ 2
 If the problem was x+2 ≥ 4, then the solution would have been (∞, 6] U [2, ∞). i.e.,
 If x + 2 > 4 ⇒ x ∈ (∞, 6) U (2, ∞)
 If x+2 ≥ 4 ⇒ x ∈ (∞, 6] U [2, ∞)
Graphing Absolute Value Inequalities
When we graph absolute value inequalities, we plot the solution of the inequalities on a graph. The image below shows how to graph linear absolute value inequalities. While graphing absolute value inequalities, we have to keep the following things in mind.
 Use open dots at the endpoints of the open intervals (i.e. the intervals like (a,b) ).
 Use closed/solid dots at the endpoints of the closed intervals (i.e. the intervals like [a,b]).
Absolute Value Inequalities Formulas
So far we have learned the procedure of solving the absolute value inequalities using the number line. This procedure works for any type of inequality. In fact, inequalities can be solved using formulas as well. To apply the formulas, first, we need to isolate the absolute value expression on the left side of the inequality. There are 4 cases to remember for solving the inequalities using the formulas. Let us assume that a is a positive real number in all the cases.
Case 1: When the Inequality Is of the Form x < a or x < a.
In this case, we use the following formulas to solve the inequality: If x < a ⇒ a < x < a, and if x < a ⇒ a < x < a.
Case 2: When the Inequality Is of the Form x > a or x > a.
In this case, we use the following formulas to solve the inequality: If x > a ⇒ x < a or x > a, and if x > a, then x < a or x > a.
Case 3: When the Inequality Is of the Form x < a or x < a
We know that the absolute value always results in a positive value. Thus x is always positive. Also, a is negative (as we assumed 'a' is positive). Thus the given two inequalities mean that "positive number is less than (or less than or equal to) negative number," which is never true. Thus, all such inequalities have no solution. If x < a or x ≤ a ⇒ No solution.
Case 4: When the Inequality Is of the Form x > a or x > a.
We know that the absolute value always results in a positive value. Thus x is always positive. Also, a is negative (as we assumed a is positive). Thus the given two inequalities mean that "positive number is greater than (or greater than or equal to) negative number," which is always true. Thus, the solution to all such inequalities is the set of all real numbers, R. x > a or x ≥ a ⇒ Set of all Real numbers, R.
Important Notes on Absolute Value Inequalities
 If parenthesis is written at a number, it means that the number is NOT included in the solution.
 If a square bracket is written at a number, it means that the number is included in the solution.
 We always use parentheses at ∞ or ∞ irrespective of the given inequality.
 We decide to use square brackets (or) parentheses for a number depending upon whether the given inequality has "=" in it.
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Absolute Value Inequalities Examples

Example 1: Solve the following absolute value inequality and write the solution in 2 forms.
 Inequality form
 Interval form
2x  1  9 ≥ 5
Solution:
The given absolute value inequality is:
2x  1  9 ≥ 5
First, we need to isolate the absolute value expression.
For that, we will add 9 on both sides. Then we get:
2x  1 ≥ 4
Using the formulas, we have
2 x1 ≤ 4 (or) 2 x1 ≥ 4
Adding 1 on both sides of the inequality,
⇒ 2 x ≤ 3 (or) 2 x ≥ 5
Dividing both sides by 2,
⇒ x ≤ 3/2 (or) x ≥ 5/2
Answer: Therefore, the solution is:
x ≤ 3/2 (or) x ≥ 5/2
(OR)
(∞, 3/2) U (5/2, ∞)

Example 2: A carpenter is making a handcrafted table. In order for the leg to fit, it needs to be 250 millimeters wide, allowing for a margin of error of 3.5 millimeters.
Write an absolute value inequality that models this relationship, and then find the range of the table leg's widths.
Solution:
Let us assume that the width of the table leg to be x.
Then the absolute value inequality that corresponds to the given scenario is,
x  250 ≤ 3.5
Using the formulas we learned:
3.5 ≤ x250 ≤ 3.5
Adding 250 on all the sides
⇒ 246.5 ≤ x ≤ 253.5
Answer: The range of the table leg's width is [246.5, 253.5] (in mm).

Example 3: Before a piece of steel can be sold for its maximum price, it must be 50 feet long with an absolute error of 2 feet.
Find the range of acceptable heights for pieces of steel that are to be sold at full price by writing an absolute value inequality to represent this situation and then solving it.
Solution:
Let us assume that the height of the piece of steel is x feet.
Then the absolute value inequality that corresponds to the given scenario is,
x  50 ≤ 2
Using the absolute value inequalities formula,
2 ≤ x50 ≤ 2
Adding 50 on all the sides
⇒ 48 ≤ x ≤ 52
Answer: The range of the acceptable heights of steels is [48, 52] in feet.
FAQs on Absolute Value Inequalities
What are Absolute Value Inequalities in Algebra?
Absolute value inequalities are inequalities in algebra that involve algebraic expressions with absolute value symbols and inequality symbols. The algebraic expressions are represented in absolute value symbol and the equals to symbol is replaced with greater than or less than symbol. The representation of absolute value inequality is ax + b  < c.
How To Solve Absolute Value Inequality?
Absolute value inequality is solved over a sequence of steps. The inequality is first considered as equality and is solved for the variable 'x'. The values of the variable are represented on the number line. Now we can identify the interval on the number line. From each of the intervals take a random number and substitute in the given inequality. The value which satisfies the inequality, and the relating interval of the value is the solution to the absolute value inequalities.
What Are The Symbols Used In Absolute Value Inequality?
The symbols used in absolute value inequalities are greater than (>), lesser than (<), greater than or equal to (>), and lesser than or equal to (<).
How To Graph Absolute Value Inequalities?
The graph of absolute value inequality is prepared by first considering it as equality. The graph of this equality is prepared, and then from this graph certain values are collected and substituted in the expression of inequality. The values which satisfy the inequality are mapped back in the graph, and that particular region of the graph represents the graph of the absolute value inequality.
Where Do We Use Absolute Value Inequalities?
The absolute value inequalities are used in reallife situations and in finding the solutions in linear programming to find the optimal solution. Many business situations require us to find the best solution and not just the solution. Here we can make use of the absolute value inequalities to represent the problem situation.
When Do You Switch the Sign in Absolute Value Inequalities?
We switch the sign in absolute value inequalities when we multiply or divide both sides of the inequality by a negative number. This implies we change the less than sign < to greater than sign > and vice versa.
What is the Difference Between Absolute Value Inequalities and Absolute Value Equalities?
Absolute Value Inequalities consist of absolute algebraic expressions with inequality signs like <, > ≤, or ≥. On the other hand, absolute value equalities consist of absolute algebraic expressions with equal to sign =.
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