Raoul is at least 3 years older than Vahn

2018 Handbook

2018 Test B Math · Question 54 of 57

Also in 2026 Practice Test B Math, question 47; 2025 Practice Test B Math, question 54.

Question

Raoul is at least 3 years older than Vahn. Which of the following inequalities gives the relationship between Raoul’s age ( $r$ ) and Vahn’s age ( $v$ )?

A $r-v\geq 3$
B $r-v\leq 3$
C $3-v\leq r$
D $3-r\leq v$

Show the answer

Correct answer: A

Raoul is at least 3 years older than Vahn, which can be written as

r ≥ v + 3

Rewrite this inequality to match the answer options:

r − v ≥ 3

Question analysis

Raoul is at least 3 years older than Vahn, and students choose the inequality that links their ages, r and v. It tests translating a phrase like "at least" into an inequality. The correct form may not match the one students write first, so they need to rearrange carefully and keep the direction of the inequality.

NYS standards: 7.EE.4 equations and inequalities from problems
6.EE.8 writing simple inequalities
Grade level: Grade 7 content, by NYS standard
Compared with grade 7: At grade 7 Writing inequalities from words is grade 6 and 7 work. The extra demand is that "at least" must become the right symbol and the choices are rearranged forms, so students must manipulate inequalities rather than just recognize one.
Estimated difficulty: Moderate: several steps, or a close reading with one strong distractor.
Thinking it demands: Depth of knowledge 2, skills and concepts · mainly understand
What makes it hard: Turning words into math major
An answer in an unexpected form moderate
A very close wrong answer moderate
Solution: 2 steps
Reading level: Question and choices: about grade 8, 28 words.
Students, last 30 days: Not enough answers yet to show (it takes 10).

Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Two skills are tested at once: translating "at least … older than" into an inequality, and recognizing that inequality after a rearrangement. The two choices that begin with 3 − v and 3 − r look like legitimate moves but say only that the two ages add up to at least 3. Testing one pair of ages that fits the sentence and one that does not, such as 20 and 12 against 13 and 12, sorts the choices without algebra.

Turning words into math major Understanding the question
"Raoul is at least 3 years older than Vahn" has to become r ≥ v + 3; "at least" calls for ≥, though the word "least" seems to point the other way. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)
An answer in an unexpected form moderate Choosing the answer
No choice is written as r ≥ v + 3, so the natural translation has to be rearranged to r − v ≥ 3 before it can be matched. (Enright, Morley & Sheehan, 2002)
A very close wrong answer moderate Choosing the answer
r − v ≤ 3 is the right r − v ≥ 3 with the symbol reversed; only deciding that "at least" means greater than or equal to separates them. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Write the sentence directly as Raoul's age compared with Vahn's age plus 3, then rearrange each choice into that form to see which one matches.

Traps

Misreads the question: Misses or misreads a condition or a word in the question: "at least", "each", "remaining", "not".
A faulty procedure: A systematic but wrong procedure: adding fractions straight across, applying an exponent to each term of a sum, dropping a sign when moving a term.
Answer in another form: The right answer is written differently from the way you worked it out.

The wrong answers, classified

B Misreads the question (Hegarty, Mayer & Monk, 1995)
Reveals a misconception Expected pull: strong
Reads "at least 3 years older" as at most 3, reversing the inequality, which would even allow Raoul to be younger than Vahn.
C A faulty procedure (Brown & Burton, 1978; Tatsuoka, 1983)
Reveals a misconception Expected pull: moderate
Moves v across r ≥ v + 3 without changing its sign, writing r ≥ 3 − v, which really says only that their ages add up to at least 3.
D A faulty procedure (Brown & Burton, 1978; Tatsuoka, 1983)
Reveals a misconception Expected pull: weak
Makes the same sign slip and also swaps the roles of r and v, giving 3 − r ≤ v, again only a statement about the sum of their ages.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Greater than or equal to: $a \geq b$; a and b are two amounts. The sign ≥ means a is greater than or equal to b, so a is b or more. The words at least mean the same thing. The sign ≤ means less than or equal to, or at most.

Sort out the question
The letter $r$ is Raoul’s age, and $v$ is Vahn’s age. Raoul is “at least 3 years older than Vahn”. That means the gap between their ages is 3 years or more. An inequality is a math sentence that uses a sign like $\geq$ instead of =. You need the one that says this.
Write the age gap
Raoul is older, so the gap is his age minus Vahn’s age. The gap is $r - v$ . For example, ages 15 and 10 have a gap of 5.
Turn at least into a sign
At least 3 means 3 or more. So use $\geq$ , which means greater than or equal to.
$r - v \geq 3$
This is choice A.
Watch out: the word least does not mean less. Choice B, $r - v \leq 3$ , says the gap is 3 or less. Then Raoul could even be younger.
Check choices C and D
Add $v$ to both sides of choice C. You get $3 \leq r + v$ . That is about the sum of their ages, not the gap. Choice D works the same way. Neither one says Raoul is older.
Give the answer
Only one choice says the gap is 3 years or more. So the answer is A, $r-v \geq 3$ .

Check your answer

Test ages that do not fit: Raoul 11 and Vahn 10. He is only 1 year older. Choice A gives $11 - 10 = 1$ , and 1 is not 3 or more. So A rightly says no. Choice B says yes, so B is wrong.

The original DOE question

Question 111 as printed on page 164 of the DOE handbook. — As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 164, question 111. Open page 164 of the handbook (PDF)