Absolute Value Inequalities. - Solving, Graph, Formula, Examples

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 step-by-step 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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• Standard Form of Linear Equations