AMC 10 · 2024 · #3

Grade 8 arithmetic

Archetype Small-Domain Constrained Search

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

📏 Short solution 💡 2 insights

Problem

For how many integer values of $x$ is $|2x| \leq 7 \pi$

$\textbf{(A) } 16 \qquad\textbf{(B) } 17 \qquad\textbf{(C) } 19 \qquad\textbf{(D) } 20 \qquad\textbf{(E) } 21$

Modified Problem in Certain China Testpapers

For how many integer values of $x$ is $|2x| \leq 6 \pi$

$\textbf{(A) } 16 \qquad\textbf{(B) } 17 \qquad\textbf{(C) } 19 \qquad\textbf{(D) } 20 \qquad\textbf{(E) } 21$

Solution for Certain China Testpapers

Use the fact that $\pi \approx 3.14$ , and thus you can get $6\pi \approx 18.84$ . We could easily see that the answer is $\{-9,-8,...,8,9\}\implies\boxed{\text{(C) }19}$

~RULE101

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: Count how many integers $x$ make the inequality $|2x| \leq 7\pi$ true.

Givens: The inequality $|2x| \leq 7\pi$; $\pi \approx 3.14159$ is irrational, slightly more than $\tfrac{22}{7}$; $x$ must be an integer (positive, negative, or zero); Answer choices: (A) $16$, (B) $17$, (C) $19$, (D) $20$, (E) $21$

Unknowns: The number of integer values of $x$ that satisfy $|2x| \leq 7\pi$

Understand

Restated: Count how many integers $x$ make the inequality $|2x| \leq 7\pi$ true.

Plan

Primary tool: #13 Convert to Algebra

Secondary: #2 Make a Systematic List

The givens are an algebraic absolute-value inequality plus an irrational bound — Tool #13 (Convert to Algebra) is the natural lead. Translate $|2x| \leq 7\pi$ into $-7\pi \leq 2x \leq 7\pi$, then divide by $2$ to get $-\tfrac{7\pi}{2} \leq x \leq \tfrac{7\pi}{2}$. Once we have a clean numerical interval, Tool #2 (Make a Systematic List) finishes the job: $\tfrac{7\pi}{2} \approx 10.996$, so the integers $-10, -9, \ldots, 10$ are exactly the ones inside the band — list them in order and count.

Execute — Answer: E

#13 Convert to Algebra 6.NS.C.7 Step 1

Unfold the absolute value.
The inequality $|2x| \leq 7\pi$ says the distance from $2x$ to $0$ is at most $7\pi$, which means $2x$ sits between $-7\pi$ and $7\pi$.

$$|2x| \leq 7\pi \;\Longleftrightarrow\; -7\pi \leq 2x \leq 7\pi$$

💡 Absolute value is just "distance from $0$ on the number line" — the Grade 6 definition that makes $|A| \leq B$ collapse into the compound inequality $-B \leq A \leq B$.

#13 Convert to Algebra 7.EE.B.4 Step 2

Solve for $x$ alone by dividing the compound inequality by the positive constant $2$, which keeps both directions of the inequality intact.

$$-\tfrac{7\pi}{2} \leq x \leq \tfrac{7\pi}{2}$$

💡 Dividing a compound inequality by a positive number is the same algebra move as for a regular equation — a Grade 7 inequality-solving step.

#13 Convert to Algebra 8.NS.A.2 Step 3

Estimate the bound numerically.
Use $\pi \approx 3.14159$, so $7\pi \approx 21.991$ and $\tfrac{7\pi}{2} \approx 10.996$.
The interval for $x$ is therefore approximately $[-10.996, 10.996]$.

$$\tfrac{7\pi}{2} \approx \tfrac{21.991}{2} \approx 10.996$$

💡 Approximating an irrational like $\pi$ with a nearby rational to pin down its size is exactly the Grade 8 "rational approximations of irrational numbers" standard.

#2 Make a Systematic List 6.NS.C.6 Step 4

List the integers inside the interval $[-10.996, 10.996]$.
Since $10.996 < 11$, the integer $11$ is excluded; since $10 < 10.996$, the integer $10$ is included.
Symmetry covers the negative side.

$$x \in \{-10, -9, -8, \ldots, -1, 0, 1, \ldots, 9, 10\}$$

💡 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 $0$.

#2 Make a Systematic List 2.OA.A.1 Step 5

Count the integers in the list.
From $-10$ up to $10$ inclusive, the count is $10 - (-10) + 1$, or equivalently $10$ negatives, $10$ positives, plus $0$.

$$10 - (-10) + 1 = 21 \;\Rightarrow\; \textbf{(E)}$$

💡 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".

[1] #13 6.NS.C.7 Unfold the absolute value. The inequality $|2x| \leq 7\pi$ says the distance fro

[2] #13 7.EE.B.4 Solve for $x$ alone by dividing the compound inequality by the positive constant

[3] #13 8.NS.A.2 Estimate the bound numerically. Use $\pi \approx 3.14159$, so $7\pi \approx 21.9

[4] #2 6.NS.C.6 List the integers inside the interval $[-10.996, 10.996]$. Since $10.996 < 11$,

[5] #2 2.OA.A.1 Count the integers in the list. From $-10$ up to $10$ inclusive, the count is $1

Review

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

Alternative: Tool #6 (Guess and Check) on the borderline integers: the question is essentially "how many integer pairs $\pm n$ satisfy $2n \leq 7\pi$, plus $x = 0$?". Test $n = 10$: $20 < 7\pi \approx 21.99$, pass. Test $n = 11$: $22 > 21.99$, fail. So allowed $n$ are $1$ through $10$, giving $10$ positives, $10$ negatives, and $0$ — total $21$.

CCSS standards used (min grade 8)

2.OA.A.1 Solve one- and two-step word problems using addition and subtraction within 100 (Counting the integers from $-10$ to $10$ using "last minus first plus one" addition.)
6.NS.C.6 Understand a rational number as a point on the number line (Picturing the interval $[-10.996, 10.996]$ on the number line so the integers inside can be listed in order.)
6.NS.C.7 Understand ordering and absolute value of rational numbers (Rewriting $|2x| \leq 7\pi$ as the compound inequality $-7\pi \leq 2x \leq 7\pi$ via the "distance from zero" interpretation.)
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities (Dividing the compound inequality by $2$ to isolate $x$ as $-\tfrac{7\pi}{2} \leq x \leq \tfrac{7\pi}{2}$.)
8.NS.A.2 Use rational approximations of irrational numbers to compare their size (Approximating $\pi \approx 3.14159$ to estimate $\tfrac{7\pi}{2} \approx 10.996$ and decide which integers fall inside the interval.)

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

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Review

CCSS standards used (min grade 8)

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