Which of the following cases must be considered when solving a one-variable absolute value inequality algebraically?

a.

the quantity inside the absolute value is non-negative

c.

the quantity inside the absolute value is negative

b.

the quantity inside the absolute value is 0

d.

both a & c

Cases a and c must be considered. If the quantity in the absolute value is non-negative, we can ignore the absolute value and solve the equation, considering only solutions that would have made the original absolute value non-negative.

If the quantity in the absolute value is negative, then we remove the absolute value sign and multiply the quantity inside by - 1. Then, we solve the new equation, considering only solutions that make the original quantity in the absolute value negative.

These cases must thus be considered. Case b is included in case a, since 0 is nonnegative. Therefore, cases a & c only must be considered. D is the best answer.