Question

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

Answer 1

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.