Question
Jordan has a budget of $120 to rent a food truck. The rental company charges $18 per hour to rent the truck, plus a one-time cleaning fee of $30.
Write an inequality that represents this situation, in which x is the number of hours Jordan can rent the truck.
Enter your answer in the space provided. Enter only your answer.
On the test you build the answer on an on-screen keypad; here, type it. Use ^ for a power, sqrt( ) or √ for a root, pi for π, and <= or >= for ≤ and ≥.
Show the answer
Correct answer: 30 + 18x ≤ 120
Let x represent the number of hours Jordan can rent the truck.
The hourly rental cost can be represented as , written as .
The one-time cleaning fee of $30 is added to the hourly cost to obtain the total cost.
The sum of the cleaning fee and the hourly rental cost is represented as .
As Jordan has $120 to spend, the total cost must be less than or equal to $120.
This is represented as or .
Question analysis
Jordan has $120 to rent a food truck that costs $18 an hour plus a one-time $30 cleaning fee, and students type an inequality for x, the number of hours. It tests writing a two-step inequality from a word problem. Matching each dollar amount to its role and choosing the symbol that fits a spending limit are the tricky parts.
- Grade level
- Grade 7 content, by NYS standard
- Compared with grade 7
- At grade 7 Writing a two-step inequality from a word problem is the grade 7 standard itself. Nothing is offered to choose from: students type the whole inequality, so the numbers, the variable term and the symbol are all theirs to get right.
- Estimated difficulty
- Moderate: several steps, or a close reading with one strong distractor.
- Thinking it demands
- Depth of knowledge 2, skills and concepts · mainly apply
- What makes it hard
- Turning words into math moderate
- An answer in an unexpected form minor
- Solution
- 3 steps
- Reading level
- Question and choices: about grade 6, 57 words.
- Corrected from the DOE handbook
- A spelling error in the explanation was corrected.
- 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
The arithmetic is small, so the points ride on two decisions the keypad leaves open: which symbol, and whether to stop before solving. A budget means Jordan may spend all $120 but no more, so the total cost must stay at or under it, and a strict < leaves out the five-hour rental that costs exactly $120. Testing x = 5 and x = 6 in the typed inequality confirms both the symbol and the setup.
Turning words into math moderate Understanding the question
The stem never says “at most”: the limit comes only from “a budget of $120”, while “$18 per hour” must multiply x and the “one-time cleaning fee of $30” is added just once. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)
An answer in an unexpected form minor Choosing the answer
The whole inequality is typed on a keypad that offers <, ≤, > and ≥ side by side, with no choices to compare, and the solved form x ≤ 5 is not accepted. (Enright, Morley & Sheehan, 2002)
Strategy: Work out what 1 hour and 2 hours would cost Jordan in all, see which amount repeats each hour, build the total for x hours, then compare it with the budget and test a few values.
Traps
- Counting the ends: Inclusive or exclusive ends, fence posts, consecutive numbers: an answer that is one too many or too few, or an endpoint drawn open where it belongs closed (or the reverse).
- Misreads the question: Misses or misreads a condition or a word in the question: "at least", "each", "remaining", "not".
The wrong answers, classified
30 + 18x < 120 Counting the ends (Brown & Burton, 1978)
Builds the cost 30 + 18x correctly but uses a strict less-than sign, which shuts out a five-hour rental costing exactly the $120 Jordan has.
30 + 18x ≥ 120 Misreads the question (Hegarty, Mayer & Monk, 1995)
Builds the cost correctly but points the symbol the wrong way, treating the $120 budget as an amount Jordan must spend at least instead of a limit.
x ≤ 5 Misreads the question (Hegarty, Mayer & Monk, 1995)
Writes and solves the inequality correctly, then enters its solution, x ≤ 5, though the item asks for the inequality showing the fee, the hourly charge and the budget.
What was corrected from the DOE handbook, and why
- Explanation: The DOE’s explanation says “The onetime cleaning fee of $30 is added to the hourly cost …”. “Onetime” means “former”; a fee charged once is a “one-time” fee, as the question itself spells it. The site now says “one-time”.
Worked solution, step by step
Formulas used
- Total cost with a fee and a rate
- The fee is a charge paid only once, so you add it once. The rate is the charge for each hour. x is the number of hours. You multiply the rate by x because it is paid again for every hour.
- Staying within a budget
- The sign ≤ means is less than or equal to. A budget is the most money you can spend. You may spend all of it, or less, but never more. So the total cost must be less than or equal to the budget.
Sort out the question
You know three facts:
- Jordan can spend at most $120. That is the budget.
- The truck costs $18 for each hour.
- A cleaning fee of $30 is paid one time only.
x is the number of hours. You need an inequality. That is a math sentence that compares two amounts with a sign such as < or ≤. It must show all three facts.
Work out a few totals
Find the total cost for a few hours. Each hour adds one more $18:
Hours Hourly charges ($) Total cost ($) 1 2 x The $30 shows up once in every total. The $18 shows up once for each hour. So x hours cost 30 + 18x dollars. Here 18x means 18 times x.
Choose the right sign
The total cost must fit inside the budget. Jordan may spend exactly $120, or less, but not more. That is less than or equal to, the sign ≤. So:
Watch out: the sign < means less than, but not equal. leaves out spending exactly $120. The sign ≥ is backward too. It says Jordan must spend at least $120. Also keep each number in its place. charges $30 an hour.
Stop before you solve
You could solve the inequality. Subtract 30 from both sides to get . Then divide both sides by 18 to get . So Jordan can rent the truck for at most 5 hours.
Watch out: the question asks for an inequality that models the situation. To model means to show the facts of the story in math. The solved form, x ≤ 5, is true, but it hides the $18, the $30 and the $120. So x ≤ 5 is not accepted. Enter the inequality before you solve it.
Enter it on the keypad
On the test, build the inequality on the keypad. Press 3, 0, +, 1, 8 and the x key. Then press the ≤ key, in the fourth row of keys. Then press 1, 2, 0. You need no dollar signs and no times sign. On this site, type it, with <= for ≤: 30 + 18x <= 120.
So you would enter 30 + 18x ≤ 120. The same inequality written the other way round, 120 ≥ 30 + 18x, is also right. The solved form, x ≤ 5, is not accepted, because it does not model the situation.
Check your answer
Test two numbers of hours. For 5 hours, . Is 120 ≤ 120? Yes, so 5 hours fit the budget. For 6 hours, . Is 138 ≤ 120? No, and $138 is over budget. The inequality gets both right.
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