Absolute Value Equations & Inequalities
Key rules for splitting absolute value problems into cases, spotting no-solution setups, and translating tolerance word problems.
Keyboard Shortcuts
💡 Pro tip: Use keyboard shortcuts for faster studying!
Study Smart Tips for Absolute Value Equations & Inequalities
Master these concepts using proven study techniques that actually work:
Active Recall
Test yourself before flipping each card to strengthen memory retention
Spaced Repetition
Review difficult cards more frequently than easy ones
Multiple Sessions
Break study time into shorter, focused sessions
Explain Aloud
Verbalize answers to reinforce understanding
Questions Covered in This Set
10 cards to master
What does |x| actually measure?
The distance of x from zero on the number line. Distance is never negative, so |x| is always ≥ 0.
Core rule: if |X| = c with c > 0, then...?
X = c or X = -c. Split into two equations and solve each one separately.
What must you do BEFORE splitting an equation like 3|2x + 1| - 4 = 11?
Isolate the absolute value bars first: 3|2x+1| = 15, then |2x+1| = 5, then split into 2x+1 = 5 and 2x+1 = -5 (x = 2 or x = -3).
When does an absolute value equation have NO solution?
When, after isolating, the absolute value equals a negative number (e.g. |x + 4| = -2). Distance can't be negative — stop, don't split.
When do you have to check for extraneous solutions?
Whenever the other side of the equation contains a variable (e.g. |x - 2| = 3x). Plug each answer back into the original and reject any that fail.
Solve |X| < c (less than). What pattern?
AND: -c < X < c. One squeezed interval around zero ('less thAND').
Solve |X| > c (greater than). What pattern?
OR: X < -c or X > c. Two rays shooting outward ('greatOR').
Solve |2x - 6| ≤ 10.
-10 ≤ 2x - 6 ≤ 10 → -4 ≤ 2x ≤ 16 → -2 ≤ x ≤ 8.
Write the tolerance pattern for word problems.
|actual - target| ≤ allowed error. Example: bottles filled to 500 mL with 8 mL tolerance → |v - 500| ≤ 8.
Which words signal ≤ (AND) versus > (OR) in tolerance problems?
'Within', 'no more than', 'at most' → ≤ (AND). 'Differs by more than' → > (OR).
