Which inequality represents the statement “a number x is at least 18”?
Hint
Does “at least 18” allow the number to be exactly 18? Can it be bigger?
Step-by-step solution
"At least 18" means the smallest allowed value is 18. So x can be 18 or more.
"18 or more" includes 18, so the symbol has the "or equal to" line: \ge.
Put the variable on the left: x \ge 18.
Answer: x \ge 18
Check:x = 18 should be allowed: 18 \ge 18 ✓. x = 25 should be allowed: 25 \ge 18 ✓. x = 10 should not: 10 \ge 18 is false ✓.
Tip: "At least" sets a floor (minimum) → \ge. "At most" sets a ceiling (maximum) → \le.
Question 2 of 36
No calculator
Movie tickets cost $12 each. Amir can spend no more than $60 on tickets. Which inequality represents the number of tickets t he can buy?
Hint
How much do t tickets cost? Then translate “no more than”.
Step-by-step solution
Variable:t = number of tickets.
Cost: $12 each, so t tickets cost 12 \times t = 12t dollars.
Limit words: "no more than $60" means $60 or less → \le 60.
Put it together:
12t \le 60
Answer: 12t \le 60
Check: 5 tickets cost 12(5) = 60, and 60 \le 60 ✓ — he can spend exactly $60. 6 tickets cost 72, and 72 \le 60 is false ✓.
Tip: A price "each" (or "per") always multiplies the variable.
Question 3 of 36
No calculatorDrop-down
A roller coaster allows only riders who are taller than 48 inches. Let h be a rider's height in inches. Choose the symbol that completes the inequality.
h48
Choose an answer from each menu.
Hint
Is a rider who is exactly 48 inches tall allowed? Which direction are the allowed heights?
Step-by-step solution
"Taller than 48 inches" means the height must be more than 48.
Is exactly 48 inches allowed? "Taller than 48" means more than 48, so a rider who is exactly 48 inches is not taller — the limit is not included. Use a strict symbol.
Bigger values are allowed → "greater than": >.
Answer: h > 48
Check: A 50-inch rider: 50 > 48 ✓ (may ride). A 48-inch rider: 48 > 48 is false ✓ (not taller than 48). A 40-inch rider: false ✓.
Tip: Ask two questions: Bigger or smaller? (direction) and Is the limit itself allowed? (strict or inclusive).
Question 4 of 36
No calculator
A café manager must spend less than $90 on bagels for a staff breakfast. Each bagel costs $1.25. Which inequality represents the number of bagels b she can buy?
Hint
Cost of the bagels = price per bagel × number of bagels. Is spending exactly $90 allowed?
Step-by-step solution
Let b = the number of bagels. Each costs $1.25, so the total cost is 1.25b dollars.
The total must be less than $90. "Less than" does not include $90 itself, so use the strict symbol <.
Put it together:
1.25b < 90
Answer: 1.25b < 90
Check: 70 bagels cost 1.25 \times 70 = \$87.50, and 87.50 < 90 ✓. 72 bagels cost exactly $90, and 90 < 90 is false ✓ — "less than $90" does not allow it.
Tip: A price per item multiplies the variable. Then decide two things: which direction (more or less) and whether the limit is included.
Question 5 of 36
No calculatorFill in
A hiker's backpack can hold at most 18 kg. Each full water bottle weighs 0.75 kg. Complete the inequality for the number of water bottles w the backpack can hold (count only the bottles).
w \le
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Which number tells the weight of one bottle, and which number is the limit?
Step-by-step solution
Let w = the number of bottles. Each weighs 0.75 kg, so w bottles weigh 0.75w kg.
The total can be at most 18 kg, which includes 18, so use \le.
The inequality is:
0.75w \le 18
So the first blank is 0.75 and the second blank is 18.
Answer: 0.75w \le 18
Check: Solving gives w \le 18 \div 0.75 = 24. Twenty-four bottles weigh 24 \times 0.75 = 18 kg — exactly the limit ✓. Twenty-five would weigh 18.75 kg ✗.
Tip: The number "per item" goes next to the variable; the limit goes alone on the other side of the symbol.
Question 6 of 36
No calculator
A greenhouse thermostat keeps the temperature Tfrom 65°F to 85°F, inclusive. Which inequality represents the temperatures the thermostat allows?
Hint
"Inclusive" tells you whether 65 and 85 themselves are allowed. The allowed temperatures are between the two numbers.
Step-by-step solution
The temperature must be at least 65°F: T \ge 65, or 65 \le T.
The temperature must be at most 85°F: T \le 85.
Both must be true at the same time, so write them together with T in the middle:
65 \le T \le 85
"Inclusive" means 65 and 85 are allowed, so both symbols are \le.
Tip: "Between" → a compound inequality with the variable in the middle and the smaller number on the left. "Inclusive" → \le; "exclusive" or "strictly between" → <.
Question 7 of 36
No calculator
Lisa uses a car-sharing service. She writes the inequality
18 + 0.25m \le 40
to find how many miles m she can drive on one trip. In this inequality, what does 0.25 represent?
Hint
Which number is multiplied by the miles? That number changes the cost every time she drives one more mile.
Step-by-step solution
Look at each part of 18 + 0.25m \le 40:
m is the number of miles.
In 0.25m, the 0.25 is multiplied by the miles, so 0.25 is the cost for each mile ($0.25 per mile).
18 is not multiplied by anything, so it is a fixed charge for the trip ($18).
40 is on the other side of \le, so it is the most she will spend ($40).
So 0.25 is the cost per mile.
Answer: The cost, in dollars, for each mile driven.
Check: If she drives 10 miles, the mileage part is 0.25 \times 10 = \$2.50 — it grows with the miles, just as a per-mile price should ✓.
Tip: In a linear inequality, the number attached to the variable is the rate (per unit); the number standing alone on the same side is the fixed amount.
Question 8 of 36
No calculator
A plumber charges $75 for a house call plus $60 per hour of work. A customer will pay no more than $300. Which inequality can be used to find the number of hours h the plumber can work?
Hint
Which amount is charged once, and which is charged for every hour?
Step-by-step solution
Variable:h = hours of work.
Rate: $60 per hour → 60h.
Fixed amount: $75 once for the house call → +75.
Limit: "no more than $300" → \le 300.
Together:
75 + 60h \le 300
Answer: 75 + 60h \le 300
Check: For 3 hours: 75 + 180 = 255 \le 300 ✓ — matches the story (3 hours costs $255). Solving gives h \le 3.75 hours.
Tip: The number attached to "per hour" goes next to h. The one-time fee stands alone.
Question 9 of 36
No calculatorDrag and drop
Drag each phrase to the symbol it means. One phrase is not used.
<
>
\le
\ge
Tap a tile, then tap a box to place it.
Hint
For each phrase, ask: bigger or smaller? And is the number itself allowed?
Step-by-step solution
Think about each phrase with the example "___ 10 people":
"fewer than" — 9 is OK, 10 is not → strictly smaller → <.
"exceeds" — 11 exceeds 10, but 10 does not exceed 10 → strictly bigger → >.
"at most" — 10 or fewer → smaller or equal → \le.
"a minimum of" — 10 or more → bigger or equal → \ge.
"is equal to" — only 10 itself → =. This is an equation, not an inequality, so it is not used.
Check: "at most 10" allows 10 ✓ (10 \le 10), but "fewer than 10" does not ✓ (10 < 10 is false).
Tip: Words like "at most", "at least", "minimum", "maximum" include the number (line under the symbol). "More than", "fewer than", "exceeds" do not.
Question 10 of 36
No calculator
Nia has $40 saved. She plans to save $18 each week. She needs at least $310 to buy a tablet. Which inequality shows the number of weeks w she must save?
Hint
Her savings after w weeks = what she has now + what she adds each week. Then translate “at least”.
Step-by-step solution
Variable:w = number of weeks.
Rate: $18 each week → 18w.
Fixed amount: $40 already saved → +40.
Limit: "at least $310" → \ge 310.
Together:
40 + 18w \ge 310
Answer: 40 + 18w \ge 310
Check: After 15 weeks: 40 + 270 = 310 \ge 310 ✓ (she can buy the tablet). After 14 weeks: 40 + 252 = 292 — not yet ✓. The inequality matches the story.
Tip: Money you already have is a fixed amount; money added every week is the rate.
Question 11 of 36
No calculatorFill in
A storage unit costs $45 per month plus a one-time $30 move-in fee. Ella can spend at most $300 in total. Complete the inequality for the number of months m she can rent the unit.
m\ +\le 300
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Which number is charged every month? Which is charged only once?
Step-by-step solution
Variable:m = number of months.
Rate: $45 per month → 45m. So the first blank is 45.
Fixed amount: $30 one time → +30. So the second blank is 30.
Limit: "at most $300" → \le 300 (already given).
45m + 30 \le 300
Answer: 45 and 30
Check: For 6 months: 45(6) + 30 = 270 + 30 = 300 \le 300 ✓ — she can rent for 6 months and spend exactly $300. For 7 months: 315 + 30 = 345, over budget ✓.
Tip: "Per month", "each", "every" → coefficient of the variable. "One-time", "fee", "already" → the constant.
Question 12 of 36
No calculatorSelect all
Select all inequalities that mean “x is no more than 10.”
Select all that apply.
Hint
“No more than 10” means 10 or less. Some choices are written with the 10 on the left — read them carefully.
Step-by-step solution
"No more than 10" means x can be 10 or less → x \le 10.
An inequality can be read from right to left if you turn the symbol with it: x \le 10 is the same as 10 \ge x ("10 is greater than or equal to x").
Check each choice:
x \le 10 ✓
10 \ge x ✓ (same statement, sides swapped)
x < 10 ✗ — leaves out 10 itself
x \ge 10 ✗ — means "at least 10"
10 \le x ✗ — means x \ge 10
Answer: x \le 10 and 10 \ge x
Check: Test x = 10 (should be allowed): 10 \le 10 ✓ and 10 \ge 10 ✓, but 10 < 10 is false. Test x = 3: 3 \le 10 ✓, 10 \ge 3 ✓, but 10 \le 3 is false.
Tip: The "open mouth" of the symbol always faces the larger amount, no matter which side the variable is on.
Question 13 of 36
No calculator
A pickup truck can haul at most 1,650 lb in its bed. A toolbox that stays in the bed weighs 120 lb. Each bag of concrete mix weighs 80 lb. Which inequality represents the number of bags b that can be loaded?
Hint
Total weight = toolbox + all the bags. Then decide whether exactly 1,650 lb is allowed.
Step-by-step solution
Variable:b = number of bags.
Rate: 80 lb each bag → 80b.
Fixed amount: the toolbox's 120 lb → +120.
Limit: "at most 1,650 lb" includes exactly 1,650 → \le 1650.
Together:
120 + 80b \le 1650
Answer: 120 + 80b \le 1650
Check: 19 bags: 120 + 1520 = 1640 \le 1650 ✓. 20 bags: 120 + 1600 = 1720 — too heavy ✓. The inequality behaves exactly like the story.
Tip: "At most" and "maximum" include the limit, so use \le, not <.
Question 14 of 36
No calculator
Paolo has a $150 gift card. He spends $12 of it on lunch each week. He wants to keep at least $30 on the card for a birthday gift. Which inequality represents the number of weeks w he can buy lunch with the card?
Hint
Does the balance go up or down each week? Write the balance after w weeks, then compare it with $30.
Step-by-step solution
The card starts with $150.
Each week the balance goes down by $12, so after w weeks it is 150 - 12w.
He wants the balance to be at least $30, which includes $30, so use \ge:
150 - 12w \ge 30
Answer: 150 - 12w \ge 30
Check: After 10 weeks: 150 - 120 = 30 \ge 30 ✓ (exactly $30 is fine). After 11 weeks: 150 - 132 = 18, which is less than $30 ✗ — so the inequality correctly stops him at 10 weeks.
Tip: Spending, using up, or draining means subtract the rate from the starting amount. Then compare the amount left with the target.
Question 15 of 36
No calculatorDrop-down
A tomato seedling is 8 cm tall and grows 1.5 cm per week. A gardener wants to know after how many weeks w the plant will be taller than 20 cm. Complete the inequality.
20
Choose an answer from each menu.
Hint
Height now plus growth over w weeks gives the height later. Is exactly 20 cm "taller than 20 cm"?
Step-by-step solution
Starting height: 8 cm. Growth: 1.5 cm each week, so 1.5w cm after w weeks.
Height after w weeks: 8 + 1.5w.
"Taller than 20 cm" means more than 20, and 20 itself is not included, so use >.
8 + 1.5w > 20
Answer: 8 + 1.5w > 20
Check: Solving: 1.5w > 12, so w > 8. After 8 weeks the plant is 8 + 12 = 20 cm — not taller than 20 ✓. After 9 weeks it is 21.5 cm ✓.
Tip: The starting amount is the number without the variable. The amount that changes each week is the number with the variable.
Question 16 of 36
No calculator
A triangular flower bed has sides of length x feet, (x + 4) feet, and 2x feet. A gardener has at most 60 feet of edging to go around the whole bed. Which inequality represents this situation?
Hint
The edging goes around the bed — that is the perimeter. Add all three sides and combine like terms.
Step-by-step solution
Edging around the bed = perimeter = sum of the three sides:
x + (x + 4) + 2x
Combine like terms: x + x + 2x = 4x, plus the 4:
4x + 4
The perimeter can be at most 60 ft:
4x + 4 \le 60
Answer: 4x + 4 \le 60
Check: Solving: 4x \le 56, so x \le 14. With x = 14 the sides are 14, 18, and 28 ft: 14 + 18 + 28 = 60 ✓.
Tip: "Around" means perimeter (add the sides). "Cover" or "inside" means area. Combine the x-terms carefully: 2x counts as twox's.
Question 17 of 36
No calculator
A parking lot charges $5 for the first hour and $3 for each additional hour. Erin will park for h hours (h \ge 1) and can pay at most $20. Which inequality represents this situation?
Hint
How many of the h hours are charged at $3? (The first hour is not one of them.)
Step-by-step solution
The first hour costs $5.
The other hours are "additional." If Erin parks h hours, that is h - 1 additional hours, costing 3(h - 1) dollars.
Total cost: 5 + 3(h - 1).
She can pay at most $20:
5 + 3(h - 1) \le 20
Answer: 5 + 3(h - 1) \le 20
Check: For 6 hours: 5 + 3(5) = 20 \le 20 ✓. For 7 hours: 5 + 3(6) = 23 ✗. The lot's rule gives the same: first hour $5 plus 5 more hours at $3 = $20 ✓.
Tip: When a price changes after the first unit (or after a threshold), the cheaper or different rate applies only to the units beyond the threshold: h - 1, t - 500, and so on.
Question 18 of 36
No calculator
The table shows what a tutor charges for sessions of different lengths. The charges follow a linear pattern.
Hours, h
1
2
3
Charge
$55
$90
$125
A family can spend at most $300 on tutoring. Which inequality represents the number of hours h they can buy?
Hint
How much does each extra hour add? What would 0 hours cost if you continued the pattern backwards?
Step-by-step solution
Rate: each extra hour adds 90 - 55 = 35 dollars (and 125 - 90 = 35 ✓).
Fixed part: go back one hour from $55: 55 - 35 = 20 dollars. So the charge is 20 + 35h.
Check with the table: 20 + 35(3) = 125 ✓.
The charge can be at most $300:
20 + 35h \le 300
Answer: 20 + 35h \le 300
Check: Solving gives 35h \le 280, so h \le 8. Eight hours cost 20 + 280 = \$300 ✓ — exactly the budget.
Tip: The first value in a table is usually not the rate. Rate = change between rows; fixed part = value when the input is 0.
Question 19 of 36
No calculator
A community theater sells adult tickets for $12 and child tickets for $7. To cover its costs, it needs ticket sales of at least $1,500. Let a be the number of adult tickets and c the number of child tickets. Which inequality represents this situation?
Hint
Find the money from adult tickets and the money from child tickets separately, then add.
Step-by-step solution
Money from adult tickets: $12 each, so 12a.
Money from child tickets: $7 each, so 7c.
Total sales: 12a + 7c.
Sales must be at least $1,500, which includes $1,500:
12a + 7c \ge 1500
Answer: 12a + 7c \ge 1500
Check: 100 adult and 50 child tickets: 1{,}200 + 350 = 1{,}550 \ge 1{,}500 ✓ — enough. 80 adult and 60 child: 960 + 420 = 1{,}380 ✗ — not enough. The inequality sorts these correctly.
Tip: With two kinds of items, write price × number for each kind and add. Make sure each price sits next to its own variable.
Question 20 of 36
No calculatorFill in
A delivery truck starts a trip with a full 42-gallon tank. It uses 0.125 gallon of fuel per mile. The driver wants at least 5 gallons left in the tank at the end. Complete the inequality for the number of miles m the truck can drive.
-m \ \ge
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Start with the fuel in the tank. What is taken away each mile? What is the least fuel allowed at the end?
Step-by-step solution
Starting fuel: 42 gallons.
Fuel used after m miles: 0.125m gallons. The tank goes down, so subtract.
Fuel left: 42 - 0.125m.
It must be at least 5 gallons:
42 - 0.125m \ge 5
The blanks are 42, 0.125, and 5.
Answer: 42 - 0.125m \ge 5
Check: Solving: -0.125m \ge -37, so m \le 296 (the sign flips). After 296 miles: 42 - 37 = 5 gallons left ✓.
Tip: In a "using up" story, the starting amount comes first, the rate is subtracted, and the amount that must remain goes on the other side.
Question 21 of 36
No calculatorFill in
A shipping company's standard rate is for packages weighing more than 2 lb and at most 70 lb. A bookstore ships books in a box that weighs 1.5 lb empty, and each book weighs 2 lb. Complete the compound inequality for the number of books b in a box that qualifies for the standard rate. Write the box weight first, then the books.
<+b \ \le
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Write the package weight as box + books. Which limit is strict ("more than") and which includes the number ("at most")?
Step-by-step solution
Package weight = box + books = 1.5 + 2b pounds.
"More than 2 lb" is strict: 2 < 1.5 + 2b.
"At most 70 lb" includes 70: 1.5 + 2b \le 70.
Both are true at once:
2 < 1.5 + 2b \le 70
The blanks are 2, 1.5, 2, and 70.
Answer: 2 < 1.5 + 2b \le 70
Check: Solving: subtract 1.5 → 0.5 < 2b \le 68.5; divide by 2 → 0.25 < b \le 34.25, so 1 to 34 books. One book: 1.5 + 2 = 3.5 lb ✓ (more than 2). 34 books: 1.5 + 68 = 69.5 lb ✓.
Tip: A compound inequality can mix symbols. Read each end separately: "more than" → < on that side; "at most" → \le on the other.
Question 22 of 36
No calculator
The length of a rectangular dog run is 5 feet more than three times its width w. The perimeter of the dog run must be less than 74 feet. Which inequality represents this situation?
Hint
Write the length in terms of w first. The perimeter of a rectangle is on the formula sheet: P = 2l + 2w.
Step-by-step solution
Length: "5 more than three times the width" → l = 3w + 5.
Perimeter (formula sheet): P = 2l + 2w.
Substitute the length:
P = 2(3w + 5) + 2w = 6w + 10 + 2w = 8w + 10
Limit: "less than 74 feet" → < 74.
8w + 10 < 74
Answer: 8w + 10 < 74
Check: If w = 4: l = 17, and the perimeter is 17 + 17 + 4 + 4 = 42; the formula gives 8(4) + 10 = 42 ✓. Solving: 8w < 64, so w < 8.
Tip: Perimeter means all four sides. Adding just one length and one width gives only half the perimeter.
Question 23 of 36
No calculator
A phone plan costs $30 per month and includes 500 text messages. Each text over 500 costs $0.05. Jordan expects to send more than 500 texts and wants his monthly bill to be at most $45. If t is the total number of texts he sends, which inequality represents this situation?
Hint
Only the texts above 500 cost extra. How do you write “the number of texts over 500”?
Step-by-step solution
Variable:t = total texts (more than 500).
Texts that cost extra: only those over 500 → t - 500.
Extra charge: $0.05 for each of those → 0.05(t - 500).
Check: If he sends 700 texts: 30 + 0.05(200) = 30 + 10 = 40 \le 45 ✓ — he pays for 200 extra texts, which matches the story. Solving gives t \le 800.
Tip: For "over", "beyond" or "after the first" thresholds, subtract the free amount from the variable inside parentheses.
Question 24 of 36
Fill in
Jada wants to buy a laptop that costs $500. She has $85 saved, but she must first pay a $45 phone bill out of her savings. She earns $14 per hour at a part-time job. Write an inequality for the number of hours h she must work so that she has at least $500 after paying the bill, and solve it.
What is the least whole number of hours she must work?
hours
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Money she will have = savings − bill + earnings. Set that greater than or equal to the laptop price.
Step-by-step solution
Money she will have after h hours: 85 - 45 + 14h.
She needs at least $500:
85 - 45 + 14h \ge 500
Simplify the left side: 85 - 45 = 40:
40 + 14h \ge 500
Tip: Money coming in is added; money going out (bills, fees) is subtracted. Combine the plain numbers first, then solve the simpler inequality.
Question 25 of 36
No calculatorSelect all
A farmer's water tank holds 250 gallons. Her animals drink 18 gallons per day. She uses the inequality
250 - 18d \ge 40
to plan when to refill the tank.
Select all statements that are true about this inequality.
Select all that apply.
Hint
Ask what each number does: Is it multiplied by the variable? Is it alone? Which side of the symbol is it on?
Step-by-step solution
250 stands alone on the left with nothing multiplied — it is the starting amount of water. True.
18 is multiplied by d and subtracted — it is the water used per day. True.
40 is on the other side of \ge — it is the least amount allowed to remain in the tank. True.
d is the number of days, not the gallons left (the gallons left are 250 - 18d). False.
\ge means "greater than or equal to," so exactly 40 gallons is allowed. False.
Answer: A, B, and C
Check: Solving: -18d \ge -210, so d \le 11.\overline{6} — she must refill within 11 full days. After 11 days: 250 - 198 = 52 gallons \ge 40 ✓.
Tip: To interpret an inequality, label every piece: start, rate, variable, symbol, limit. Words like "per day" always belong to the number next to the variable.
Question 26 of 36
No calculator
The graph shows the charge left on Omar's phone battery while he uses it.
Omar wants to stop using the phone while the battery still has at least 20% charge. Which inequality represents the number of hours t he can use the phone?
Hint
From the graph, find the starting charge and how much it drops each hour. Is the line going up or down?
Step-by-step solution
Starting value: at t = 0 the charge is 100%.
Rate: from (0, 100) to (5, 60) the charge drops 40 points in 5 hours: 40 \div 5 = 8 points per hour. The line falls, so the rate is negative.
Charge after t hours: 100 - 8t.
It must stay at least 20%:
100 - 8t \ge 20
Answer: 100 - 8t \ge 20
Check: The graph passes through (10, 20). Solving: -8t \ge -80, so t \le 10 ✓ — matches the graph.
Tip: A falling line means subtract the rate from the starting value. Always confirm your rate with a second labeled point.
Question 27 of 36
No calculatorDrop-down
A farmer plans to plant corn and soybeans. Seed and fertilizer cost $120 per acre of corn and $90 per acre of soybeans. She can spend no more than $24,000. Let c be acres of corn and s be acres of soybeans.
24{,}000
Choose an answer from each menu.
Hint
Write cost per acre × acres for each crop and add. Then choose the symbol that means no more than.
Step-by-step solution
Corn costs $120 per acre: 120c.
Soybeans cost $90 per acre: 90s.
Total cost: 120c + 90s.
"No more than" $24,000 means the total can equal $24,000 or be less: \le.
120c + 90s \le 24000
Answer: 120c + 90s \le 24{,}000
Check: 100 acres of corn and 120 of soybeans: 12{,}000 + 10{,}800 = 22{,}800 \le 24{,}000 ✓. 150 acres of corn and 100 of soybeans: 18{,}000 + 9{,}000 = 27{,}000 ✗ — over budget.
Tip: In a two-variable inequality, each rate stays glued to its own variable. Never add the two rates together unless both items are the same.
Question 28 of 36
Fill in
A rectangular patio will be 12 ft wide. Its length L must give an area of at least 300 square feet, and the border trim around the patio (the perimeter) can be at most 80 feet. Write an inequality for each condition and solve them. Complete the range of possible lengths.
\le L \le
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Area of a rectangle = length × width. Perimeter = 2 × length + 2 × width. Write one inequality for each, then combine.
Step-by-step solution
Area condition:A = 12L, and it must be at least 300:
12L \ge 300 \;\Rightarrow\; L \ge 25
Perimeter condition:P = 2L + 2(12) = 2L + 24, and it must be at most 80:
2L + 24 \le 80 \;\Rightarrow\; 2L \le 56 \;\Rightarrow\; L \le 28
Both conditions must hold:
25 \le L \le 28
Answer: 25 \le L \le 28
Check:L = 25: area = 300 ✓, perimeter = 74 \le 80 ✓. L = 28: area = 336 \ge 300 ✓, perimeter = 80 ✓. L = 29: perimeter = 82 ✗.
Tip: Two conditions on the same quantity often give a lower limit and an upper limit. Put them together as one compound inequality.
Question 29 of 36
An online store is having a 15% off sale on everything. After the discount, it adds a flat $6 shipping charge. Mia can spend at most $80 in total. Let x be the total of the regular prices of the items she buys. Which inequality represents this situation?
Hint
After a 15% discount, what percent of the regular price does she pay? Is the shipping discounted?
Step-by-step solution
A 15% discount means she pays 100\% - 15\% = 85\% of the regular price: 0.85x.
The shipping is added after the discount, so it is not discounted: 0.85x + 6.
The total can be at most $80:
0.85x + 6 \le 80
Answer: 0.85x + 6 \le 80
Check: Solving: 0.85x \le 74, so x \le 87.06. Items with regular prices totaling $87: 0.85(87) + 6 = 73.95 + 6 = 79.95 \le 80 ✓.
Tip: "p\% off" → multiply by (1 - p/100). Pay attention to what is discounted — fees added afterwards stay at full price.
Question 30 of 36
A youth soccer league needs at least $5,000 for new equipment. It already has $1,250. It will hold a raffle, selling tickets for $5 each, but printing the tickets costs $150. Which inequality models the number of tickets t it must sell, and what is its solution?
Hint
Money the league will have = money already raised + ticket sales − printing cost.
Step-by-step solution
Money the league will have: 1250 + 5t - 150.
It needs at least $5,000:
1250 + 5t - 150 \ge 5000
Tip: Costs are subtracted, money already in hand is added. Write every amount into the inequality before you solve.
Question 31 of 36
No calculator
A climbing gym charges a monthly membership fee plus a fee for each visit. The table shows the total monthly cost for some numbers of visits v.
Visits, v
0
4
8
Monthly cost
$40
$64
$88
Rhea wants her monthly cost to be less than $130. Which inequality represents this situation?
Hint
The cost at 0 visits is the membership fee. How much do 4 more visits add?
Step-by-step solution
Fixed fee: at 0 visits the cost is $40.
Rate: 4 more visits add 64 - 40 = 24 dollars, so each visit costs 24 \div 4 = 6 dollars.
Cost: 40 + 6v. (Check: 40 + 6(8) = 88 ✓.)
The cost must be less than $130 — strict, so 130 is not allowed:
40 + 6v < 130
Answer: 40 + 6v < 130
Check: Solving: 6v < 90, so v < 15. 14 visits cost 40 + 84 = \$124 < \$130 ✓; 15 visits cost exactly $130, which is not less than $130 ✓.
Tip: Divide the change in cost by the change in visits — the table may skip numbers, so the rate is not always the difference between neighboring cells.
Question 32 of 36
Fill in
A drama club sells programs for $2.50 each. It has already sold 60 cupcakes at $1.50 each, and it spent $65 on printing. The club wants a total profit of at least $300.
Write an inequality for the number of programs p the club must sell, and solve it. What is the least number of programs the club must sell?
programs
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Profit = all money coming in − money spent. Money in comes from programs and cupcakes.
Tip: Write profit as (money in) − (money out) before you add the limit. Forgetting the cupcake money gives 146, and forgetting the printing cost gives 84.
Question 33 of 36
A doctor orders a medicine dose of 4 mg to 6 mg per kilogram of body weight, inclusive. The patient weighs 70 kg. The medicine comes in 50-mg tablets. Which inequality represents the number of tablets n that give a dose in the ordered range?
Hint
First find the smallest and largest total doses in mg for a 70-kg patient. Then write the total from n tablets.
Step-by-step solution
Smallest total dose: 4 \times 70 = 280 mg.
Largest total dose: 6 \times 70 = 420 mg.
n tablets of 50 mg give 50n mg.
The dose must be between 280 and 420 mg, inclusive:
280 \le 50n \le 420
Tip: "Per kilogram" means multiply by the body weight before writing the range. Then the amount from n units goes in the middle.
Question 34 of 36
Drop-down
Marcus rents a moving truck. He writes and solves this inequality for the number of miles m he can drive:
35 + 0.40m \le 60 \qquad\Longrightarrow\qquad m \le 62.5
In this situation, the solution means Marcus can drive , and the number 35 represents .
Choose an answer from each menu.
Hint
What does the symbol \le say about 62.5? Which number in the inequality is not multiplied by m?
Step-by-step solution
The solution.m \le 62.5 means the miles must be less than or equal to 62.5. So Marcus can drive at most 62.5 miles without going over his limit.
The 35. In 35 + 0.40m, the 0.40 multiplies m, so 0.40 is the cost per mile. The 35 is not multiplied by m, so it is the fixed rental fee he pays no matter how far he drives.
The 60 on the right is his budget of $60.
Answer: at most 62.5 miles; the fixed rental fee ($35)
Check: At 62.5 miles: 35 + 0.40(62.5) = 35 + 25 = \$60 — exactly the budget ✓. At 70 miles: $63 ✗.
Tip: Read a solution out loud in words: "m \le 62.5" → "miles at most 62.5." Then check that the meaning matches the story.
Question 35 of 36
Select all
A moving van can carry at most 1,500 lb of furniture. Each desk weighs 60 lb and each chair weighs 15 lb. The load is modeled by 60d + 15c \le 1500, where d is desks and c is chairs.
Select all loads the van can carry.
Select all that apply.
Hint
Substitute each pair into 60d + 15c and compare with 1,500. Exactly 1,500 is allowed.
Step-by-step solution
Compute 60d + 15c for each load and compare with 1,500:
Answer: 18 desks and 25 chairs; 10 desks and 60 chairs; 25 desks and 0 chairs
Check: The failing loads are only 15 lb and 45 lb over — close, which is why each one must actually be computed rather than guessed.
Tip: A solution of a two-variable inequality is a pair of numbers. Substitute both, keep each weight with its own item, and remember \le includes the boundary.
Question 36 of 36
No calculator
Maria works two part-time jobs. She earns $16 per hour at a store and $22 per hour tutoring. She must earn at least $600 per week, and she can work at most 35 hours per week in total. Let s be her store hours and t her tutoring hours. Which system of inequalities represents this situation?
Hint
Write one inequality for money and one for time. Which numbers go with money, and which with hours?
Step-by-step solution
Money: store pay 16s plus tutoring pay 22t, and she must earn at least $600:
16s + 22t \ge 600
Time: total hours s + t, and she can work at most 35:
s + t \le 35
Both must be true together.
Answer: 16s + 22t \ge 600 and s + t \le 35
Check: 15 store hours and 18 tutoring hours: pay = 240 + 396 = 636 \ge 600 ✓ and time = 33 \le 35 ✓ — a schedule that works. 30 store hours and 5 tutoring hours: pay = 480 + 110 = 590 ✗.
Tip: In a system, each inequality describes one limit. Check units: dollars with dollars, hours with hours.
