Absolute Value Inequalities

Introduction :

Normally absolute value means it is nothing but the values without considering it sign. Here we are going to solve the absolute values for inequalities. If we solve the absolute value inequalities we will get two values. Using this we value we will find the limitations of the inequalities. We will see some example problems for absolute value inequalities. Normally absolute value of |a| = `+-` a

Problems Based on Absolute Values Inequalities.

Example 1 for absolute value inequalities:

Find the absolute values of the inequalities. |x - 2| >= 10

Solution:

Given inequality is |x - 2| `gt= ` 10

To find the absolute value for the inequality is

(x – 2) `gt=` 10 ………… (1) And –(x – 2) `gt=` 10 …………………. (2)

Equation 1:

(x – 2) `gt= ` 10 ………… (1)

Add +2 on both sides

x - 2 + 2 `gt=` 10 +2

x `gt=` 12

Equation 2:

-(x – 2) `gt=` 10

-x + 2 `gt=` 10

Add -2 on both sides.

-x + 2 – 2 `gt=` 10 – 2

-x `gt=` 8

Divide by -1.

x `lt=` -8

So x lies between -8 `gt=` x `gt=` 12

We will see some more example problems based on absolute value inequalities.


Example 2 for Absolute Value Inequalities:

Find the absolute values of the inequalities. |x - 5| >= 2

Solution:

Given inequality is |x - 5| `gt=` 2

To find the absolute value for the inequality is

(x – 5) `gt= ` 2 ………… (1) And –(x – 5) `gt= ` 2 …………………. (2)

Equation 1:

(x – 5) `gt= ` 2 ………… (1)

Add +5 on both sides

x - 5 + 5 `gt= ` 2 + 5

x `gt= ` 7

Equation 2:

-(x – 5) `gt= ` 2

-x + 5 `gt= ` 2

Add -5 on both sides.

-x + 5 - 5 `gt= ` 2 - 5

-x `gt= ` -3

Divide by -1.

x `lt= ` 1

So x lies between 7 `gt= ` x `gt= ` 1                                            

These are some of the examples for absolute value inequalities. Here we will treat the inequalities like absolute values. Normally the difference between the inequalities and linear equation is less than and greater than sign.