# Absolute Value in Algebra

**Absolute value** in algebra refers to the distance of a number from zero on the number line, without considering its direction.

The absolute value of a number x is written as |x| and is equal to x if x is positive, and is equal to -x if x is negative.

In this article, we will discuss absolute value and properties of absolute value.

**Properties of Absolute Value**

The absolute value of a number has the following properties:

- |x| â‰¥ 0 for all real numbers x. The absolute value is always zero or positive.
- |x| = x if x â‰¥ 0. The absolute value of a positive number or zero is equal to the number itself.
- |x| = -x if x < 0. The absolute value of a negative number is equal to the number without its negative sign.
- |-x| = |x|. The absolute value of the opposite of x is the same as the absolute value of x.
- |x + y| â‰¤ |x| + |y|. The absolute value of a sum is less than or equal to the sum of the absolute values.

**Absolute Value Equations**

An absolute value equation is an equation that contains an absolute value expression.

For example,

|x – 5| = 3

To solve an absolute value equation, we need to consider two cases:

- x – 5 = 3, which gives x = 8
- x – 5 = -3, which gives x = 2

The two solutions are x = 8 and x = 2.

**Absolute Value Inequalities**

Absolute value inequalities are inequalities that contain absolute value expressions.

For example,

|x – 3| < 5

To solve this, we again consider two cases:

- x – 3 < 5, which gives -2 < x < 8
- x – 3 > -5, which gives -8 < x < 2

Combining these two ranges gives us the solution set:

-8 < x < 8

**Solved Examples Absolute Value**

**Example**

Solve |x + 2| = 7

**Solution **

x + 2 = 7 or x + 2 = -7

x = 5 or x = -9

**Example**

Solve |2x – 1| > 5

**Solution**

2x – 1 > 5 or 2x – 1 < -5

2x > 6 or 2x < -4

x > 3 or x < -2

So the solution is x < -2 or x > 3

**Example**

If |x| = 4, find the possible values of x.

**Solution**

If |x| = 4, then x can either be 4 or -4.

The possible values of x are 4 and -4.

**FAQs**

**What is the absolute value of a number?**

The absolute value of a number is its distance from zero on the number line, without considering direction. For any real number x, the absolute value is written as |x| and is equal to x if x is positive, and equal to -x if x is negative.

**How do you solve an absolute value equation?**

To solve an absolute value equation, isolate the absolute value expression and set it equal to two expressions, one positive and one negative. Solve each equation separately and combine the solutions.

**How do you solve an absolute value inequality?**

To solve an absolute value inequality, isolate the absolute value expression and write two inequalities, one with a positive and one with a negative expression. Solve each inequality separately and combine the ranges.

** What are the properties of absolute value?**

The main properties are:

1) |x| â‰¥ 0 for all real x

2) |x| = x if x â‰¥ 0

3) |x| = -x if x < 0

4) |-x| = |x|

5) |x + y| â‰¤ |x| + |y|

**What happens to the absolute value of a number when you multiply it by -1?**

Multiplying a number by -1 flips it across the origin on the number line. However, since absolute value only considers distance from zero, |-x| = |x|. Multiplying by -1 does not affect the absolute value.

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