**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.

**Related Articles**

- Domain and Range
- Linear Equations
- Standard Form of Linear Equations

## 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 real-life 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 =.

## FAQs

### How do you solve an absolute value inequality by graphing? ›

Value reverse the inequality symbol and change the sign of the seven. So now we'll solve these and

**How do you solve absolute value inequalities? ›**

Absolute Value Inequalities. Here are the steps to follow when solving absolute value inequalities: Isolate the absolute value expression on the left side of the inequality. If the number on the other side of the inequality sign is negative, your equation either has no solution or all real numbers as solutions.

**What is an example of an absolute value inequality? ›**

For example, the expression **|x + 3| > 1** is an absolute value inequality containing a greater than symbol. There are four different inequality symbols to choose from. These are less than (<), greater than (>), less than or equal (≤), and greater than or equal (≥).

**How do you find the equation of an absolute value graph? ›**

Now this graph you should recognize as an absolute value graph for absolute value graphs your

**How do you solve and graph absolute value inequalities on a number line? ›**

Solving Absolute Value Equations and Inequalities - YouTube

**How do you write an inequality from a graph? ›**

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**How do you write an absolute value inequality with a given solution? ›**

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**How do you solve an absolute value inequality with two variables? ›**

Absolute Value Inequalities in Two Variables - Examples - YouTube

**How do you find the absolute value? ›**

To find the absolute value of any real number, first locate the number on the real line. **The absolute value of the number is defined as its distance from the origin**. For example, to find the absolute value of 7, locate 7 on the real line and then find its distance from the origin.

**What is absolute value function with example? ›**

For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point. An absolute value equation is **an equation in which the unknown variable appears in absolute value bars**. For example, |x|=4,|2x−1|=3,|5x+2|−4=9.

### What is the absolute value of a graph? ›

The absolute value graph **depicts the distance of a number from the origin**. The graph of the absolute value function is symmetric about the y -axis. The graph of the absolute value function makes a right angle at the origin. Absolute value function is an even function because f(x)=f(−x) f ( x ) = f ( − x ) .

**How do you write an equation given a graph? ›**

To find the equation of a graphed line, **find the y-intercept and the slope in order to write the equation in y-intercept (y=mx+b) form**. Slope is the change in y over the change in x.

**How do you write and solve an inequality? ›**

Write and Solve an Inequality to Represent a Situation - YouTube

**What is an inequality for a graph? ›**

Graphing inequalities is **the process of showing what part of the number line contains values that will "satisfy" the given inequality**. Examine the first inequality x > -5. A graph of this inequality will show what numbers may be used to replace x in our inequality to make a true statement.

**How do you write the equation of an inequality? ›**

Equations and inequalities are both mathematical sentences formed by relating two expressions to each other. In an equation, the two expressions are deemed equal which is shown by the symbol =. Where as in an inequality, the two expressions are not necessarily equal which is indicated by the symbols: **>, <, ≤ or ≥**.

**How do you write an absolute value equation from a word problem? ›**

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**How do you rewrite an absolute value equation? ›**

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**How do you graph an absolute value inequality and shade? ›**

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**How do you simplify absolute value? ›**

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**What are some examples of absolute value? ›**

Absolute Value Examples and Equations

**|6| = 6** means “the absolute value of 6 is 6.” |–6| = 6 means “the absolute value of –6 is 6.” |–2 – x| means “the absolute value of the expression –2 minus x.” –|x| means “the negative of the absolute value of x.”

### What is the absolute value of 4 by 5? ›

The absolute value of 4/5is **4/5** only ...

**What is an absolute value in math? ›**

Absolute Values. The absolute value of a number refers to the distance of a number from the origin of a number line.

**How do you graph an absolute value inequality on a TI 84? ›**

How to Graph Absolute Value Equations and Inequalities on the TI ...

**How do you graph solutions to two variables in absolute value inequalities? ›**

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**How do you solve absolute value inequalities with variables on both sides? ›**

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**How do you solve an inequality with 2 absolute values? ›**

Two Ways to Solve an Absolute Inequality with Two Absolute Values

**How do you do absolute value on a calculator? ›**

TI Calculator Tutorial: Absolute Value - YouTube

**How do you find the absolute value? ›**

To find the absolute value of any real number, first locate the number on the real line. **The absolute value of the number is defined as its distance from the origin**. For example, to find the absolute value of 7, locate 7 on the real line and then find its distance from the origin.

**How do you graph absolute value on a TI 83 Plus? ›**

Absolute Value Graphs on TI 83/84 - YouTube

**How do you graph and shade an absolute value inequality? ›**

Use a dotted or solid line, depending on the inequality sign, to connect the points. After graphing using a table of values, shade in the appropriate region. If y is greater than the absolute value quantity, then shade above the graph. If y is less than the absolute value quantity, then shade below the graph.

### How do you simplify absolute value expressions with variables? ›

Simplify Absolute Value Expressions - YouTube

**How do you solve 3 step inequalities? ›**

How to Solve Multi-Step Inequalities - YouTube

**How do you solve inequalities with variables on one side? ›**

To solve an inequality in one variable, **first change it to an equation (a mathematical sentence with an “=” sign) and then solve.** **Place the solution, called a “boundary point”, on a number line**. This point separates the number line into two regions.