When an absolute value occurs with an inequality, there are two basic situations which occur:

| x | < a            and         | x | > a.

For both cases, we can derive the formula using the definition of  absolute value:

Case 1:

| x | < a

- x < a    if   x < 0 ,  or  x < a   if  x 0.

This gives us :

x > -a    if    x < 0   , or    x < a  if x 0.

-a < x <0 ,      or     0 x < a.

This means that :

-a < x < a.

Case 2:

| x | > a

-x > a     if   x < 0 , or  x > a   if  x 0.

Similarly

x < -a   if   x < 0 , or   x > a    if  x 0.

This gives us:

x < -a        or     x  >  a.

Example 1:

| x - 5 | < 3

Using the result from Case 1,

-3 < x - 5 < 3

Adding 5 to both sides, we get our answer:

2 < x < 8.

Or in Interval Notation:

( 2 , 8 ).

Example 2:

| x - 5 | > 3

Using the result from Case @,

x - 5 < -3     or     x - 5 > 3

and,

x < 2   or   x > 8

Or in Interval Notation:

( , 2 ) U ( 8 , ).

Summary

| x | < a      gives      -a < x < a.

| x | > a      gives        x < -a        or     x  >  a.

Absolute Value

Interval Notation

Inequalities

Advanced Inequalities

Back to FAQ's