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Toolkit + CCSS Solution
Understand
Restated: The expression 2x/3 - x/6 turns out to be a whole number. Given that fact, decide which single description of x is forced to be true.
Givens: The value of $\frac{2x}{3}-\frac{x}{6}$ is an integer.; x stands for a whole number being described by the five choices.
Unknowns: Which property of x (negative, even, multiple of 3, multiple of 6, or multiple of 12) must always hold.
Understand
Restated: The expression 2x/3 - x/6 turns out to be a whole number. Given that fact, decide which single description of x is forced to be true.
Givens: The value of $\frac{2x}{3}-\frac{x}{6}$ is an integer.; x stands for a whole number being described by the five choices.
Plan
Primary tool: #7 Identify Subproblems
Secondary: #6 Guess and Check, #3 Eliminate Possibilities
The messy-looking expression hides a much simpler one. Split the task into two smaller problems: first simplify $\frac{2x}{3}-\frac{x}{6}$ to a single fraction, then ask what that fraction being an integer demands of x. Once the demand is clear, test a value to see which extra properties are NOT forced, and eliminate the choices that claim too much.
Execute — Answer: B
#7 Identify Subproblems 7.EE.A.1Step 1
Combine into one fraction
Put both terms over the common denominator 6.
Since $\frac{2x}{3}=\frac{4x}{6}$, subtracting gives $\frac{4x}{6}-\frac{x}{6}=\frac{3x}{6}$, which reduces to $\frac{x}{2}$.
💡 One well-chosen example can prove that a claimed rule is not actually required.
#3 Eliminate Possibilities 4.OA.B.4Step 4
Eliminate and conclude
Even is required, so cross out any choice that demands more.
Multiple of 3, multiple of 6, and multiple of 12 all fail the $x=2$ test, and negative fails too.
What remains is exactly 'even, but not necessarily a multiple of 3.' The answer is (B).
$$x\text{ even},\ x\text{ not necessarily a multiple of }3$$
💡 When one property survives every test and the stronger ones do not, that surviving property is the answer.
[1]
#7 7.EE.A.1Put both terms over the common denominator 6. Since $\frac{2x}{3}=\frac{4x}{6}$,
[2]
#7 4.OA.B.4The expression equals $\frac{x}{2}$, so saying it is an integer is the same as s
[3]
#6 4.OA.B.4Try $x=2$. Then $\frac{x}{2}=1$, an integer, so $x=2$ is allowed. But 2 is not a
[4]
#3 4.OA.B.4Even is required, so cross out any choice that demands more. Multiple of 3, mult
Review
Reasonableness: Check both directions. If x is even, write $x=2k$; then $\frac{x}{2}=k$ is an integer, so every even x works. If x is odd, $\frac{x}{2}$ ends in .5 and is never an integer. So the allowed x are precisely the even numbers - no more, no less - which matches choice (B).
Alternative: Instead of testing a value, reason directly: $\frac{x}{2}$ integer forces $x=2k$. Nothing in $x=2k$ pins down divisibility by 3, so multiples of 3, 6, and 12 are not required, leaving only 'even' as the guaranteed property.
CCSS standards used (min grade 7)
7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions (Combining $\frac{2x}{3}-\frac{x}{6}$, a linear expression with rational coefficients, into the single term $\frac{x}{2}$.)
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite (Recognizing that $\frac{x}{2}$ is an integer exactly when x is a multiple of 2, and that x need not be a multiple of 3, 6, or 12.)
⭐ Simplify the messy expression first: 2x/3 - x/6 is really just x/2, so x only has to be even.
⭐ Simplify the messy expression first: 2x/3 - x/6 is really just x/2, so x only has to be even.