Level Up: Beyond Linear
If you completed Set 1, you've built a foundation in translating linear equations and systems. Now it gets harder.
These 20 problems cover inequalities, quadratics, exponential models, rates of work, and multi-step translations — the categories where SAT test-takers lose the most points to setup errors.
⚠️Common mistake: On inequality problems, students frequently flip the direction. Watch for "at most" (use ≤), "at least" (use ≥), and "cannot exceed" (use ≤). The wording is designed to trip you up.
Same rules: read the problem, write only the equation or inequality, target under 30 seconds. Don't solve.
1. Inequality Direction Traps
1
A school bus can hold at most 48 passengers. There are already 17 students on the bus, and the bus will make 3 more stops. The driver's seat is separate from the passenger count. If each stop picks up students, write an inequality for the number of students that can be picked up at each stop.
2
A cell phone plan includes 5 GB of data. Each app update uses 0.15 GB, and background sync uses a flat 0.8 GB per month regardless of usage. The plan was activated on a Tuesday. If the user cannot exceed the data limit, write an inequality for the number of app updates allowed.
3
A warehouse requires that the weight on any shelf be no more than 200 pounds. Each box weighs 12 pounds, and the shelf already has a 35-pound machine on it. The warehouse has 8 shelves. Write an inequality for the number of boxes that can be placed on the shelf.
4
To qualify for free shipping, a customer must spend at least . The customer has already added items worth to their cart. Each additional item costs . The website shows 142 items in stock. Write an inequality for the price per additional item if the customer adds exactly 3 more items to qualify for free shipping.
2. Areas, Products & Parabolas
5
A rectangular garden is 3 feet longer than it is wide. A 2-foot-wide gravel path surrounds the garden, but the path area is not included in the garden's area. If the area of the garden alone is 180 square feet, write an equation to find the width of the garden.
6
A ball is thrown upward from a platform 20 feet above the ground with an initial velocity of 48 feet per second. The platform is on a building with 6 floors. Write an equation for the height of the ball after seconds.
7
The product of two consecutive even integers is 288. Write an equation to find the smaller integer.
8
A farmer wants to enclose a rectangular area using 100 feet of fencing along three sides (the fourth side is a barn wall that is 60 feet long, though the full length isn't used). If the area enclosed must be 1,200 square feet, and the side parallel to the barn has length , write an equation in terms of .
9
A company's profit in thousands of dollars is modeled by the number of units sold: the profit equals the revenue per unit (which is 50 minus the number of units, due to bulk discounts) times the number of units, minus a fixed cost of . The company employs 23 people. Write the equation for profit.
3. Doubling, Halving & Compounding
10
A town's population is 25,000 and doubles every 15 years. The town was founded in 1850. Write an equation for the population after years from now.
11
A car purchased for depreciates at a rate of 18% per year. The car has 4 doors and gets 28 miles per gallon. Write an equation for the car's value after years.
12
A bank account earns 4.5% interest compounded annually. The initial deposit is . The account holder is 34 years old. Write an equation for the account balance after years.
13
A radioactive substance has a half-life of 8 hours. A lab starts with 500 grams of the substance at 9:00 AM on a Monday. The lab is on the 4th floor. Write an equation for the amount remaining after hours.
4. Who Finishes First?
14
Pipe A can fill a tank in 6 hours. Pipe B can fill the same tank in 10 hours. The tank holds 3,000 gallons. Write an equation for how long it takes both pipes working together to fill the tank.
15
A boat travels 24 miles upstream and 24 miles downstream. The boat's speed in still water is 10 mph, and the current flows at mph. The river is 150 feet wide. If the total round trip takes 5.2 hours, write an equation to find the speed of the current.
16
Worker A can complete a project in 12 days. Worker B can complete it in 18 days. They work together for days, then Worker A finishes the remaining work alone. The project involves 3 phases, but all phases are identical in workload. If the total time is 9 days, write an equation.
5. Boss Level
17
A store sells notebooks for each and pens for each. A student buys some of each and spends exactly before tax. The store also sells erasers for . Tax is 8% on the total purchase. Write an equation for the total amount paid including tax.
18
A rectangular pool is being filled by a hose that delivers water at 15 gallons per minute. The pool is 20 feet long, 12 feet wide, and 5 feet deep. One cubic foot holds 7.48 gallons. The pool cover weighs 30 pounds. Write an equation for the time in minutes to fill the pool.
19
A phone battery drains at 5% per hour during active use and 1% per hour on standby. The phone was manufactured in 2023. If the battery starts at 100% and is used actively for hours and on standby for hours throughout a 16-hour day until it reaches 20%, write the equation.
20
An investment of dollars is split between two accounts: one earning 6% simple annual interest and the other earning 3.5% simple annual interest. The investor's broker charges a flat annual fee, which is separate from the interest. If the total interest earned in one year (before the fee) is , and dollars is placed in the 6% account, write an equation.
40 problems. 10 categories. You crushed it.
That's serious reps — most students never practice translation this way. You're ahead of the curve.