Question

Explain why there are either two, one, or no solutions for absolute value equatins. Demonstrate an example for each

Answer 1

No solution: because the absolute value is positive and can't equal a negative

|x| = -1          

One solution: because zero only has one sign

|x| = 0      ⇒ x = 0      

Two solutions: absolute value is positive whether x is positive or negative.

|x| = 2     ⇒ x = 2 or x = -2      

Answer 2

Absolute value equations |a|=b have two solutions only if b>0, because it means that a=b or a=-b. Example:
$|x + 1| = 1 \\ x = 0 \: or \: x = - 2$
Absolute value equations |a|=b have one solution only if b=0, because 0=-0. Example:
$|x + 1| = 0 \\ x = - 1$
Absolute value equations |a|=b have no solution only if b<0, because absolute value can't have a negative value. Example:
$|x + 1| = - 1 \\ x \in \varnothing$