10 core problems covering linear equations, inequalities, systems, and functions.
0 of 10 answered
01Linear EquationsCommon Mistake
📘 Example First
Anna earns $12 per hour. After working h hours she has $84. Equation: \(12h = 84\), so \(h = 7\).
Marcus rents a bike for a flat fee of $5 plus $3 per hour. He pays a total of $20. Which equation correctly represents this situation, and how many hours did he rent the bike?
⚠️ Watch out: students often forget the flat fee and only write 3h = 20.
⚡ Memory Key
FLAT + RATE × TIME = TOTAL
Correct: D — The equation is 3h + 5 = 20. Subtract 5: \(3h = 15\), divide: \(h = 5\). The flat fee ($5) is added to the rate ($3) times hours. Choice A forgets the flat fee. Choice B swaps flat fee and rate. Choice C subtracts instead of adding the flat fee.
02InequalitiesSign Flip!
📘 Example First
Solve \(-2x < 8\): divide both sides by −2 and flip the sign → \(x > -4\).
A school needs to order at least 150 chairs. They already have 62. Chairs come in boxes of 8. What is the minimum number of boxes they must order?
⚠️ Tricky: always round UP for "at least" problems — never round down.
⚡ Memory Key
AT LEAST → ≥ → ROUND UP
Correct: C (11 boxes) — Need: \(150 - 62 = 88\) more chairs. Boxes needed: \(\frac{88}{8} = 11\) exactly. So 11 boxes. Note: if the division didn't come out even, you'd round UP.
03Systems of EquationsHigh Frequency
📘 Example First
Two numbers add to 10, one is 4 more than the other. Let \(x + y = 10\) and \(x = y + 4\). Substitute: \((y+4)+y=10 \Rightarrow y=3, x=7\).
Two friends together have $47. One friend has $9 more than the other. How much does each friend have?
⚠️ Classic trap: students add $9 to the total instead of setting up a system.
⚡ Memory Key
SUM + DIFFERENCE → SUBSTITUTION
Correct: B — $19 and $28 — Let \(x + y = 47\) and \(x = y + 9\). Substitute: \((y+9)+y=47 \Rightarrow 2y=38 \Rightarrow y=19, x=28\). Check: \(28-19=9\) ✓ and \(28+19=47\) ✓.
04Percent & ProportionConfusing
A jacket originally costs $80. It is on sale for 25% off, and then there is an additional 10% off the sale price. What is the final price?
⚠️ Big trap: 25% + 10% ≠ 35% off. Each discount is applied separately!
⚡ Memory Key
SEQUENTIAL DISCOUNT: MULTIPLY FACTORS, NOT ADD %
Correct: C — $54.00 — Step 1: \(80 \times 0.75 = 60\). Step 2: \(60 \times 0.90 = 54\). Combined multiplier: \(0.75 \times 0.90 = 0.675\), so total discount is 32.5%, NOT 35%. Choice A applies 35% off incorrectly.
05Rate & DistanceMust Know
📘 Example First
\(d = r \times t\). If speed = 60 mph and time = 2.5 hrs, then distance = \(60 \times 2.5 = 150\) miles.
Train A leaves a station at 60 mph. Train B leaves the same station 1 hour later going the same direction at 80 mph. After how many hours (from Train B's departure) will Train B catch Train A?
⚠️ Students often forget Train A has a 1-hour head start (= 60 miles).
⚡ Memory Key
CATCH UP: HEAD-START ÷ SPEED DIFFERENCE
Correct: D — 3 hours — Train A's head start: \(60 \times 1 = 60\) miles. Speed gap: \(80 - 60 = 20\) mph. Time to close 60 miles at 20 mph: \(\frac{60}{20} = 3\) hours. Verify: Train A travels \(60 \times 4 = 240\) miles total; Train B travels \(80 \times 3 = 240\) miles ✓.
06Slope & Linear FunctionsConfusing
A candle is 12 inches tall and burns at a rate of 0.5 inches per hour. Which function gives the candle's height \(h\) after \(t\) hours, and after how many hours will it be completely burned?
⚡ Memory Key
h = START − RATE × t (decreasing = subtract)
Correct: B — Function: \(h = 12 - 0.5t\). Set \(h = 0\): \(12 - 0.5t = 0 \Rightarrow t = 24\) hours. The slope is negative because height decreases. Choice A has the wrong sign on the slope.
07Age ProblemsMost Missed
Lisa is 3 times as old as her brother now. In 4 years, she will be twice as old as her brother. How old is each person now?
⚠️ Remember: "In 4 years" means ADD 4 to BOTH people's ages.
⚡ Memory Key
FUTURE AGE = NOW + YEARS LATER (both sides!)
Correct: C — Let brother = \(b\), Lisa = \(3b\). In 4 years: \(3b+4 = 2(b+4)\). Expand: \(3b+4=2b+8 \Rightarrow b=4\). Lisa = 12. Check: now \(12 = 3 \times 4\) ✓; future \(16 = 2 \times 8\) ✓.
08Mixture ProblemsExam Favorite
How many liters of a 40% acid solution must be mixed with 6 liters of a 10% acid solution to get a 20% acid solution?
⚡ Memory Key
CONCENTRATION × VOLUME = AMOUNT OF SOLUTE
Correct: B — 3 liters — Let \(x\) = liters of 40% solution. Equation: \(0.40x + 0.10(6) = 0.20(x+6)\). Expand: \(0.40x + 0.60 = 0.20x + 1.20\). Simplify: \(0.20x = 0.60 \Rightarrow x = 3\).
09Word → InequalityDirection Trap
A store requires a minimum purchase of $50 for free shipping. Sam has already spent $31.75. He sees items costing $4.25 each. What is the fewest number of items he must buy to get free shipping?
⚡ Memory Key
MINIMUM → ≥ → CEILING (round up)
Correct: C — 5 items — Need: \(31.75 + 4.25n \geq 50\). So \(4.25n \geq 18.25\), \(n \geq 4.29...\). Since \(n\) must be a whole number, round up: \(n = 5\). Check: \(31.75 + 4.25(5) = 31.75 + 21.25 = 53 \geq 50\) ✓.
10Functions & DomainConceptual
A parking garage charges $3 for the first hour and $1.50 for each additional hour. The total cost function is \(C(h) = 3 + 1.5(h-1)\) for \(h \geq 1\). How much does it cost to park for 5 hours? What does \(h = 0\) represent in this context, and is it in the domain?
⚡ Memory Key
DOMAIN = REAL-WORLD VALID INPUTS ONLY
Correct: B — \(C(5) = 3 + 1.5(5-1) = 3 + 1.5(4) = 3 + 6 = \$9.00\). \(h=0\) means 0 hours parked — the car isn't there, so this has no real-world meaning and is NOT in the domain. Domain is \(h \geq 1\) (at least 1 hour).
Part 2
Geometry — Core Problems
10 essential problems: angles, triangles, circles, area, and coordinate geometry.
0 of 10 answered
01Parallel Lines & AnglesAlways Mixed Up
📘 Example First
When a transversal cuts two parallel lines: alternate interior angles are equal; co-interior (same-side) angles are supplementary (add to 180°).
Two parallel lines are cut by a transversal. One co-interior (same-side interior) angle is \((3x + 15)°\) and the other is \((2x + 5)°\). Find \(x\).
⚠️ Co-interior angles ADD to 180°, they are NOT equal. This is the #1 confusion!
⚡ Memory Key
CO-INTERIOR = SUPPLEMENTARY (C shape = 180°)
Correct: C — \(x = 32\) — Co-interior angles sum to 180°: \((3x+15)+(2x+5)=180\). Combine: \(5x+20=180 \Rightarrow 5x=160 \Rightarrow x=32\). Check: \(3(32)+15=111°\) and \(2(32)+5=69°\); \(111+69=180°\) ✓.
02Triangle Interior AnglesFundamental
In triangle \(ABC\), angle \(A = (2x)°\), angle \(B = (x + 30)°\), and angle \(C = (x - 6)°\). Find the measure of each angle.
\(a^2 + b^2 = c^2\) where \(c\) is always the hypotenuse (longest side, opposite the right angle).
A 13-foot ladder leans against a wall. The bottom of the ladder is 5 feet from the wall. How high up the wall does the ladder reach?
⚠️ The ladder is the hypotenuse (c = 13), NOT one of the legs!
⚡ Memory Key
HYPOTENUSE = LONGEST / SLANTED SIDE
Correct: C — 12 feet — \(a^2 + 5^2 = 13^2\). So \(a^2 = 169 - 25 = 144\). \(a = \sqrt{144} = 12\) feet. This is the 5-12-13 Pythagorean triple — worth memorizing!
04Circle: Area & CircumferenceDiameter vs Radius
A circular pizza has a diameter of 16 inches. What is its area? (Leave answer in terms of \(\pi\).)
⚠️ The formula uses RADIUS, not diameter. If given diameter, divide by 2 first!
⚡ Memory Key
A = πr² (radius!), C = πd (diameter ok)
Correct: B — \(64\pi\) sq in — Radius \(r = 16 \div 2 = 8\). Area \(= \pi r^2 = \pi(8)^2 = 64\pi\). Choice A uses diameter directly in the formula (\(16^2\pi\)) — the classic mistake.
05Similar TrianglesProportion Power
Triangle \(ABC\) is similar to triangle \(DEF\). Side \(AB = 6\), \(BC = 9\), and the corresponding side \(DE = 10\). What is the length of \(EF\)?
⚡ Memory Key
SIMILAR = SAME SHAPE → SET UP PROPORTION
Correct: C — 15 — Set up proportion: \(\frac{AB}{DE} = \frac{BC}{EF} \Rightarrow \frac{6}{10} = \frac{9}{EF}\). Cross-multiply: \(6 \cdot EF = 90 \Rightarrow EF = 15\).
06Exterior Angle TheoremSneaky Theorem
In a triangle, an exterior angle measures \((7x + 5)°\). The two non-adjacent interior angles are \((3x + 10)°\) and \((2x + 15)°\). Find \(x\).
⚠️ Exterior angle = SUM of the two non-adjacent interior angles (not just one!).
A rectangle is 10 cm × 6 cm. A semicircle with diameter equal to the width (6 cm) is cut out from one end. What is the remaining area? (Use \(\pi \approx 3.14\).)
⚠️ Remember: diameter = 6, so radius = 3. Semicircle area = half of circle area.
In a 30-60-90 triangle: sides are in ratio \(1 : \sqrt{3} : 2\) (short leg : long leg : hypotenuse).
In a 30-60-90 triangle, the hypotenuse is 14. What are the lengths of the two legs?
⚠️ Students often confuse which leg goes with which angle — the side opposite 30° is the SHORT leg.
⚡ Memory Key
30-60-90 → x, x√3, 2x (hyp = 2 × short leg)
Correct: B — Hypotenuse \(= 2x = 14 \Rightarrow x = 7\). Short leg (opposite 30°) \(= 7\). Long leg (opposite 60°) \(= 7\sqrt{3} \approx 12.12\). Choice C uses \(\sqrt{2}\) — that's for 45-45-90 triangles!