AMC 10 · 2024 · #3

Grade 8 arithmetic

Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In

interval-arithmeticestimationsystematic-enumeration convert-to-algebrabound-inequality-then-enumerate ↑ Prerequisites: interval-arithmeticorder-of-operationsestimation

📏 Short solution 💡 2 insights

Problem

Count how many integers x make the inequality |2x| ≤ 7π true.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Convert to Algebra

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 - $\frac{7π}{2}$ ≤ x ≤ $\frac{7π}{2}$ . Once we have a clean numerical interval, Tool #2 (Make a Systematic List) finishes the job: $\frac{7π}{2}$ ≈ 10.996, so the integers -10, -9, …, 10 are exactly the ones inside the band — list them in order and count.

1STEP 1

Absolute value is distance from 0, so |2x| ≤ 7π means 2x lies between -7π and 7π.

|2x| ≤ 7π ⟺ -7π ≤ 2x ≤ 7π

2STEP 2

Divide every part by the positive 2 to isolate x, giving - $\frac{7π}{2}$ ≤ x ≤ $\frac{7π}{2}$ .

\frac{7π}{2}

≤ x ≤

\frac{7π}{2}

3STEP 3

With π ≈ 3.14, the bound $\frac{7π}{2}$ ≈ 10.996, so x lies in about [-10.996, 10.996].

\frac{7π}{2}

≈

\frac{21.991}{2}

≈ 10.996

4STEP 4

Since 10.996 ＜ 11, the integers that fit are -10 through 10, symmetric about 0.

x ∈ {-10, -9, -8, …, -1, 0, 1, …, 9, 10}

5STEP 5

Count -10 up to 10 inclusive: 10 - (-10) + 1 integers — 10 negatives, 10 positives, plus 0.

10 - (-10) + 1 = 21 → (E)

Answer

Boundary check: does x = 11 really fail? |2 · 11| = 22 and 7π ≈ 21.99 ＜ 22, so yes 11 fails. Does x = 10 really pass? |2 · 10| = 20 and 20 ＜ 21.99, so yes 10 passes. The interval [-10, 10] has 21 integers, matching (E).

💡Key takeaway

This AMC 10 problem only needs Grade 8 rational approximations of π (plus Grade 6 absolute value) — once you know $\frac{7π}{2}$ ≈ 10.996, you just count -10 through 10 on the number line!

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