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| ≥ 0 for all real numbers x. The absolute value is always zero or positive.
• |x| = x if x ≥ 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| ≤ |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:

1. x – 5 = 3, which gives x = 8
2. 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:

1. x – 3 < 5, which gives -2 < x < 8
2. 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