AMC 10 · 2024 · #3
Grade 8 arithmeticPick an answer.
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
It unfolds into a two-sided bound.
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 is a distance from zero, so a bound on it becomes a two-sided range.
▸ Why?
A number and its opposite are the same distance from zero, so the range reaches equally far both ways.
▸ Why?
Being within that distance is exactly two comparisons at once, one on each side.
Divide by two
The bound becomes seven pi over two.
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 as a decimal
About 10.996, just short of eleven.
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
Everything from negative ten up to ten qualifies.
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 them
That is 21 integers.
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 12 problem only needs Grade 8 rational approximations of π (plus Grade 6 absolute value) — once you know 7π/2 ≈ 10.996, you just count -10 through 10 on the number line!