Question

How do you solve and graph `abs(x-8)=11`?

Answer 1

Use the definition of the absolute value function:

`A = {(A;A>=0),(-A;A < 0):}`

Separate into two equations.
Simplify the domain restrictions.
Solve the equations.

Explanation:

Given: `abs(x-8)=11`

Use the definition:

`{(x-8=11;x-8>=0),(-(x-8)=11;x-8 < 0):}`

Separate into two equations:

`x-8=11;x-8>=0 and -(x-8)=11;x-8 < 0`

Simplify the domain restrictions:

`x-8=11;x>= 8 and -(x-8)=11;x < 8`

Multiply the second equation by -1:

`x-8=11;x>= 8 and x-8=-11;x < 8`

Add 8 to both sides of both equations:

`x=19;x>= 8 and x=-3;x < 8`

Because the equations do not violate the restrictions we can drop the restrictions:

`x=19 and x=-3 larr` the answers