Cheat Sheet – Absolute Value Equations and Inequalities
Isolate the absolute value before you do anything else. Once the absolute value is alone on one
side of the equal sign, the following apply:
No Solution:
1.
2.
3.
4.
Absolute Value = negative number
Absolute Value < negative
Absolute Value ≤ negative
Absolute Value < 0
No solution (absolute can’t = a negative)
No solution (absolute can’t be < a negative)
No solution (absolute can’t be ≤ a negative)
No solution (absolute can’t be < 0)
One Solution:
1. Absolute Value = 0
2. Absolute Value ≤ 0
Expect one answer - solve “stuffing = 0”
Expect one solution (treat as Absolute Value = 0)
Two Solutions:
1. Absolute Value = positive number Expect two answers
a. Stuffing = number
b. Stuffing = – number
2. Absolute Value < positive number Expect two solutions
a. Stuffing – flip arrow – negative number (stuffing-flip-neg)
b. Stuffing – copy arrow – positive number
(stuffing-copy-copy)
3. Absolute Value ≤ positive number Expect two solutions
a. Stuffing – flip arrow – negative number (stuffing-flip-neg)
b. Stuffing – copy arrow – positive number
(stuffing-copy-copy)
4. Absolute Value > positive number Expect two solutions
a. Stuffing – flip arrow – negative number (stuffing-flip-neg)
b. Stuffing – copy arrow – positive number
(stuffing-copy-copy)
5. Absolute Value ≥ positive number Expect two solutions
a. Stuffing – flip arrow – negative number (stuffing-flip-neg)
b. Stuffing – copy arrow – positive number
(stuffing-copy-copy)
6. Absolute Value > 0
All real numbers EXCEPT where absolute = zero
All Real Numbers:
1. Absolute Value > negative
2. Absolute Value ≥ negative
3. Absolute Value ≥ 0
All real numbers (positive always > negative)
All real numbers (positive always ≥ negative)
All real numbers (positive always ≥ zero)