Competition · AMC preparation · step 4 of 4
AMC 10 · 2024B · #3
Grade 8 arithmeticPick an answer.
AMC 10 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The givens are an algebraic absolute-value inequality plus an irrational bound — Tool #13 (Convert to Algebra) is the natural lead. Translate |2x| ≤ 7π into -7π ≤ 2x ≤ 7π, then divide by 2 to get -7π/2 ≤ x ≤ 7π/2. Once we have a clean numerical interval, Tool #2 (Make a Systematic List) finishes the job: 7π/2 ≈ 10.996, so the integers -10, -9, …, 10 are exactly the ones inside the band — list them in order and count.
Unfold the absolute value
Absolute value is distance from 0, so |2x| ≤ 7π means 2x lies between -7π and 7π.
Absolute value is just "distance from 0 on the number line" — the Grade 6 definition that makes |A| ≤ B collapse into the compound inequality -B ≤ A ≤ B.
An absolute value bound collapses into a pair of ordinary bounds on either side.
▸ Why?
A quantity and its opposite sit the same distance from zero, so the sign is dropped.
▸ Why?
So being within a bound means sitting between the two opposite ends, and nothing outside qualifies.
Divide both sides by 2
Divide every part by the positive 2 to isolate x, giving - ≤ x ≤ .
Dividing a compound inequality by a positive number is the same algebra move as for a regular equation — a Grade 7 inequality-solving step.
7.EE.B.4Convert To AlgebraEstimate the bound
With π ≈ 3.14, the bound ≈ 10.996, so x lies in about [-10.996, 10.996].
Approximating an irrational like π with a nearby rational to pin down its size is exactly the Grade 8 "rational approximations of irrational numbers" standard.
8.NS.A.2Convert To AlgebraList the integers inside
Since 10.996 ＜ 11, the integers that fit are -10 through 10, symmetric about 0.
Walking along the number line and listing each integer that sits inside a given interval is the Grade 6 "number line" idea, applied symmetrically around 0.
6.NS.C.6Make A Systematic ListCount the integers
Count -10 up to 10 inclusive: 10 - (-10) + 1 integers — 10 negatives, 10 positives, plus 0.
Counting how many whole steps it takes to walk from one end of a list to the other is a Grade 2 add/subtract word-problem skill — "last minus first plus one".
2.OA.A.1Make A Systematic ListThis AMC 10 problem only needs Grade 8 rational approximations of π (plus Grade 6 absolute value) — once you know ≈ 10.996, you just count -10 through 10 on the number line!
- Unfold the absolute value
- Divide both sides by 2
- Estimate the bound
- List the integers inside
- Count the integers
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