Step 1 of 10
Start
Find the hard limit hidden in the words.
Meaning first
A room holds at most 80 people.
Two drinks cost $2 each; popcorn costs $3 per bag. Spend at most $18.
Trail ride: $50 fixed + $15 per hour, with at most $80.
Spot cap
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Build cost
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Solve
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Check
Number line · locate the cap
A filled circle includes the limit
Shading shows every allowed value. A filled endpoint means the boundary value counts too.
Room capacity number lineAT MOST 80
all shaded values are allowed
79shaded
80filled = included
81unshaded
79 is shadedless than 80 → allowed
● 80 is filledequal to 80 → included
81 is unshadedgreater than 80 → blocked
79 is allowed because it is shaded. 80 is allowed because its circle is filled.
81 is outside the shaded region, so it is not allowed.
people ≤ 80
The endpoint tells you whether 80 counts
x < 80open circle · 80 excluded
x ≤ 80filled circle · 80 included
Build one total before the cap
Fixed cost and repeating cost belong on the same side. The spending cap belongs on the other side.
Madeline buys exactly two $2 drinks and some $3 popcorn bags. Her total spending cannot be more than $18.
cannot be more than $18→≤ 18
2 × $2 = $4fixed drink cost
$3 × pp popcorn bags
$18hard spending cap
$0spending railCAP $18
$4
$3
$3
$3
$3
$2 left
A fifth bag would make $19 and cross the cap.+$3 → $19
Add every cost into one total
4+3p≤18
fixedrepeatingcap
Free the budget, then count whole bags
The symbol stays ≤ because we subtract and divide by positive numbers.
4 + 3p ≤ 18
− 4 from both sides
3p ≤ 14
÷ 3 dollars per bag
p ≤ 14 ÷ 3 = 4.66...
Popcorn bags are whole objects, so the greatest allowed whole number is 4.
4 + 3(4) = 164 bags stay under $18 ✓
4 + 3(5) = 195 bags cross the cap ✕
Your turn: model the trail ride
Use one variable h for hours. Build the cost, choose the cap symbol, then give the maximum whole hours.
Fill: fixed cost, hourly rate, cap symbol, cap amount, and greatest whole hours.
✎
Solve on your whiteboard
Let h be hours. Write the cost inequality, solve it, and test the boundary. Submit when your work is ready.
Work submitted. The cap receipt is unlocked.
The cap receipt proves 2 hours
Each line preserves the same hard limit from the story.
Trail ride constraint receipt
readfixed 50 · rate 15 · cap 80
Add every cost into one total
model50 + 15h ≤ 80
− 50 from both sides
remaining15h ≤ 30
÷ 15 dollars per hour
maximumh ≤ 2 hours
boundary50 + 15(2) = 80 ✓
past cap50 + 15(3) = 95 ✕
Reliable model: fixed amount + rate × number of units ≤ the cap.
Cap language mastered
At most, no more than, cannot exceed, and up to all point to the same hard-stop model.
≤limit included
Read from the story: put every changing cost on one side, the hard cap on the other, and include equality at the boundary.
Spot cap wordsBuild one totalSolve the boundaryTest the next value