AMC 10 · 2010 · #13
Grade 8 algebraPick an answer.
AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
An absolute value is really two rules in disguise: |A| = A when A ≥ 0 and |A| = -A when A ＜ 0. With one absolute value inside another, Tool #7 (Identify Subproblems) splits the tangle into a few clean linear equations, one per sign region. Tool #3 (Eliminate Possibilities) then throws out any answer that lands outside the region it was built for, so only the genuine solutions survive to be added.
The value must be non-negative
The outer absolute value can never be negative, so every solution has to satisfy x ≥ 0.
Distance from zero is never negative, so x cannot be negative.
A distance from zero is never negative, so the unknown cannot be negative here.
▸ Why?
An absolute value measures size and drops the sign, exactly as a square does.
▸ Why?
Anything at or above zero can never equal something below it, so negatives are ruled out at once.
Split on the inner absolute value
The inner |60 - 2x| changes rule at x = 30. For x ≤ 30 it equals 60 - 2x, so the equation becomes x = |4x - 60|.
Peeling one absolute value at a time turns a scary nested expression into an ordinary one.
6.NS.C.7Identify SubproblemsSolve the first case
Split |4x - 60| at x = 15: x ≥ 15 gives x = 4x - 60, so 20; below 15 gives x = 60 - 4x, so 12.
Each sign choice gives a plain linear equation you can solve in one line.
8.EE.C.7Identify SubproblemsSolve the second case
For x greater than 30 the inside is 2x - 60, the two 2x terms cancel, and the equation gives x = |60| = 60.
In this range the two 2x terms cancel, leaving a constant that answers itself.
8.EE.C.7Identify SubproblemsCollect and add the solutions
All three candidates 12, 20 and 60 sit inside the range they came from, and their sum is 92, choice (C).
Keep only the values that satisfy the original equation, then add them for the final answer.
6.EE.B.5Eliminate PossibilitiesPeel a nested absolute value one layer at a time: each sign choice becomes a simple line equation, and you keep only the answers that land in their own range.
- The value must be non-negative
- Split on the inner absolute value
- Solve the first case
- Solve the second case
- Collect and add the solutions