AMC 10 · 2021 · #1

Grade 7 arithmetic

Archetype Small-Domain Constrained Search

absolute-valuesystematic-enumerationestimationinterval-arithmetic systematic-enumerationbound-inequality-then-enumerate ↑ Prerequisites: absolute-value

📏 Short solution 💡 2 insights

Problem

How many integer values of $x$ satisfy $|x|<3\pi$ ?

Pick an answer.

(A)

(B)

~10

(C)

~18

(D)

~19

(E)

~20

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Toolkit + CCSS Solution

Understand

Restated: Count the integers $x$ whose distance from $0$ on the number line is less than $3\pi$.

Givens: $|x| < 3\pi$, where $\pi \approx 3.14159$; $x$ must be an integer; Answer choices: (A) $9$, (B) $10$, (C) $18$, (D) $19$, (E) $20$

Unknowns: The number of integers $x$ satisfying the inequality

Understand

Restated: Count the integers $x$ whose distance from $0$ on the number line is less than $3\pi$.

Givens: $|x| < 3\pi$, where $\pi \approx 3.14159$; $x$ must be an integer; Answer choices: (A) $9$, (B) $10$, (C) $18$, (D) $19$, (E) $20$

Plan

Primary tool: #1 Draw a Diagram

Secondary: #2 Make a Systematic List, #3 Eliminate Possibilities

Tool #1 (Diagram) is perfect — sketch a number line with $-3\pi$ and $3\pi$ as endpoints, then mark integer tick marks inside. Tool #2 (Systematic List) handles the counting once we know where the boundary integers are. Tool #3 (Eliminate) checks the answer choices: counting symmetric integers around $0$ always gives an odd total (positives $+$ negatives $+$ zero), so (C) $18$ and (E) $20$ can be eliminated immediately, leaving (A) $9$, (B) $10$, or (D) $19$.

Execute — Answer: D

#1 Draw a Diagram 7.NS.A.1 Step 1

Rewrite the absolute value inequality as a range.
$|x| < 3\pi$ means $x$ sits within $3\pi$ units of zero — so $-3\pi < x < 3\pi$.

$$|x| < 3\pi \;\Longleftrightarrow\; -3\pi < x < 3\pi$$

💡 Absolute value is distance from $0$ — Grade 7 "rational number distance on the number line".

#1 Draw a Diagram 7.G.B.4 Step 2

Estimate $3\pi$ to find the boundary integers.
Since $\pi \approx 3.14$, $3\pi \approx 9.42$.
So we need integers strictly between $-9.42$ and $9.42$.

$$3\pi \approx 3 \times 3.14 = 9.42$$

💡 Knowing $\pi \approx 3.14$ is the Grade 7 circle-formula standard — used here just for size.

#2 Make a Systematic List 6.NS.C.7 Step 3

List the integers strictly between $-9.42$ and $9.42$.
The largest is $9$ (since $9 < 9.42$ and $10 > 9.42$) and the smallest is $-9$.
Write them in order: $-9, -8, \dots, 0, \dots, 8, 9$.

$$\{-9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$$

💡 Ordering integers on the number line — Grade 6 standard.

#2 Make a Systematic List 4.NBT.B.4 Step 4

Count the listed integers.
There are $9$ negatives, $9$ positives, and $0$ — total $9 + 9 + 1 = 19$.
That matches choice (D).

$$9 + 9 + 1 = 19 \;\Rightarrow\; \textbf{(D)}$$

💡 Add three small whole numbers — Grade 4 standard algorithm.

#3 Eliminate Possibilities 4.OA.C.5 Step 5

Sanity check by Eliminate.
Any symmetric range around $0$ gives an odd count (matching pairs plus zero), so (C) $18$ and (E) $20$ are impossible.
$19$ is the only odd choice in the right ballpark, confirming (D).

$$\text{odd count} \;\Rightarrow\; \textbf{(D)}$$

💡 Even/odd parity argument — Grade 4 "generate and analyze patterns".

[1] #1 7.NS.A.1 Rewrite the absolute value inequality as a range. $|x| < 3\pi$ means $x$ sits wi

[2] #1 7.G.B.4 Estimate $3\pi$ to find the boundary integers. Since $\pi \approx 3.14$, $3\pi \

[3] #2 6.NS.C.7 List the integers strictly between $-9.42$ and $9.42$. The largest is $9$ (since

[4] #2 4.NBT.B.4 Count the listed integers. There are $9$ negatives, $9$ positives, and $0$ — tot

[5] #3 4.OA.C.5 Sanity check by Eliminate. Any symmetric range around $0$ gives an odd count (ma

Review

Reasonableness: $3\pi \approx 9.42$, so the range $(-9.42, 9.42)$ definitely includes $-9$ through $9$ and excludes $-10, 10$. The count of $19$ is between (C) $18$ and (E) $20$, exactly the odd value that symmetry around $0$ forces.

Alternative: Tool #13 (Convert to Algebra) — solve $-3\pi < x < 3\pi$ directly, take floor and ceiling to get $-9 \le x \le 9$, then use the formula $9 - (-9) + 1 = 19$. Same answer.

CCSS standards used (min grade 7)

4.OA.C.5 Generate a number or shape pattern following a given rule (Spotting that integer counts symmetric around zero are always odd, eliminating choices (C) and (E).)
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers (Adding $9 + 9 + 1 = 19$ at the end.)
6.NS.C.7 Understand ordering and absolute value of rational numbers (Listing the integers between $-9.42$ and $9.42$ in numerical order.)
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers (Reading $|x| < 3\pi$ as distance from $0$ and rewriting it as $-3\pi < x < 3\pi$.)
7.G.B.4 Know the formulas for area and circumference of a circle (Using $\pi \approx 3.14$ to estimate $3\pi \approx 9.42$.)

⭐ This AMC 10 problem only needs Grade 7 "absolute value means distance from zero" and knowing $\pi \approx 3.14$ — sketch the number line from $-9.42$ to $9.42$, list the integers inside, and count $19$!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 7)

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