How do you know if an absolute value equation has no solution?

Absolute value functions are always greater than or equal to zero.

Answer: There are 0 solutions to the equation |-5 - 4x| + 9 = -40

There is one major red flag of an equation with an absolute value that has no solution.

Explanation:

An absolute value expression can never be less than zero. That is a fully reduced absolute value expression must be greater than or equal to zero.

If on simplifying an absolute value expression, the value on the other side of the equation has a negative sign i.e. it is a negative number then it has no solutions.

Lets verify this with an example,

⇒ |-5 - 4x| + 9 = -40

First, we have to isolate the absolute value expression, so we'll subtract 9 from both sides.

⇒ |-5 - 4x| = -49

We observe that LHS is positive while RHS is negative, which contradicts.

Since the number on the right-hand side comes out to be negative, thus the absolute function has 0 solutions possible.