Set up: 120 + 15h = 285
Subtract 120: 15h = 165
Divide by 15: h = 11
✅ Maya needs 11 hours. Trap: some students use $285 directly without subtracting what's already saved.
3
Solving Inequalities
Solve and express the solution set: −3x + 7 > 16
KEY: FLIP the inequality sign when multiplying/dividing by NEGATIVE
📐 Step-by-Step Explanation
Subtract 7: −3x > 9
Divide by −3 → FLIP the sign: x < −3
✅ The #1 mistake: forgetting to flip the inequality when dividing by a negative number!
4
Slope & Linear Functions
Line ℓ passes through the points (2, −1) and (6, 7). What is the slope-intercept form of the line?
Slope formula: m = (y₂ − y₁) / (x₂ − x₁)
KEY: SLOPE = RISE over RUN → then plug in one point to find b
📐 Step-by-Step Explanation
Slope: m = (7 − (−1))/(6 − 2) = 8/4 = 2
Use point (2, −1): −1 = 2(2) + b → −1 = 4 + b → b = −5
Answer: y = 2x − 5
✅ Trap: watch the double negative when subtracting y₁ = −1.
5
Systems of Equations · Word Problem
Two friends, Alex and Ben, are saving money. Alex has $30 and saves $10/week. Ben has $90 and saves $4/week. After how many weeks will they have the same amount?
KEY: SET the two equations EQUAL → solve for the variable
📐 Step-by-Step Explanation
Alex: A = 30 + 10w, Ben: B = 90 + 4w
Set equal: 30 + 10w = 90 + 4w
6w = 60 → w = 10 weeks
Check: Alex = 30+100 = $130, Ben = 90+40 = $130 ✓
6
Exponent Rules
Simplify: (x3)2 · x4
KEY: POWER of POWER = multiply exponents · SAME BASE multiply = add exponents
📐 Step-by-Step Explanation
(x³)² = x3×2 = x⁶
x⁶ · x⁴ = x6+4 = x¹⁰
✅ Trap: some students multiply 3×2×4 = 24. Remember: power rule first, then product rule (add).
7
Quadratic Equations · Factoring
Solve: x2 − 5x − 14 = 0
KEY: Find two numbers that MULTIPLY to c and ADD to b → (x + p)(x + q)
📐 Step-by-Step Explanation
Need two numbers: multiply = −14, add = −5
Those numbers: +7 and −2 (7 × −2 = −14, 7 + (−2) = +5)... wait!
We need −5: try −7 and +2: (−7)(+2)=−14 ✓, −7+2=−5 ✓
So: (x − 7)(x + 2) = 0 → x = 7 or x = −2
8
Word Problem · Consecutive Integers
The sum of three consecutive even integers is 78. What is the largest of the three integers?
KEY: CONSECUTIVE EVEN = n, n+2, n+4 (not n, n+1, n+2!)
📐 Step-by-Step Explanation
Let integers be: n, n+2, n+4
n + (n+2) + (n+4) = 78 → 3n + 6 = 78 → 3n = 72 → n = 24
Largest = 24 + 4 = 28
✅ Trap: students find n = 24 and stop — but the question asks for the largest integer!
9
Functions & Function Notation
If f(x) = 3x2 − 2x + 1, what is f(−2)?
KEY: SUBSTITUTE −2 everywhere you see x → use PARENTHESES around negative values!
📐 Step-by-Step Explanation
f(−2) = 3(−2)² − 2(−2) + 1
= 3(4) − (−4) + 1
= 12 + 4 + 1 = 17
✅ Trap: (−2)² = +4, NOT −4. The square makes it positive!
10
Mixture / Percent Word Problem
A store sells two types of coffee: Brand A at $8/lb and Brand B at $12/lb. A manager creates a 20-pound blend worth $9.40/lb. How many pounds of Brand A are used?
KEY: VALUE equation → 8x + 12(20−x) = 9.40 × 20
📐 Step-by-Step Explanation
Let x = lbs of Brand A → (20 − x) = lbs of Brand B
8x + 12(20−x) = 9.40 × 20
8x + 240 − 12x = 188
−4x = −52 → x = 13 pounds
Check: 8(13) + 12(7) = 104 + 84 = 188 = 9.4 × 20 ✓
A right triangle has legs of length 9 and 12. What is the length of the hypotenuse?
a² + b² = c² → hypotenuse = c (the longest side)
KEY: LEGS go in as a and b · HYPOTENUSE is always c (opposite right angle)
📐 Step-by-Step Explanation
9² + 12² = c² → 81 + 144 = c² → c² = 225 → c = √225 = 15
✅ This is the classic 3-4-5 triple scaled by 3: (9, 12, 15). Recognize Pythagorean triples to save time!
13
Area & Perimeter · Word Problem
A rectangular garden has a perimeter of 56 meters. The length is 4 meters more than twice the width. What is the area of the garden?
KEY: PERIMETER = 2L + 2W · AREA = L × W
📐 Step-by-Step Explanation
Let W = width, L = 2W + 4
2(2W+4) + 2W = 56 → 4W + 8 + 2W = 56 → 6W = 48 → W = 8 m
L = 2(8) + 4 = 20 m
Check perimeter: 2(20) + 2(8) = 40 + 16 = 56 ✓
Area = L × W = 20 × 8 = 160 m²
✅ Trap C: using W=9 or miscalculating. Always verify perimeter before finding area!
14
Circle — Area & Circumference
A circle has a diameter of 10 cm. What is its area? (Use π ≈ 3.14)
Area = πr² | Circumference = 2πr | r = diameter ÷ 2
KEY: AREA uses RADIUS (not diameter!) → always halve the diameter first
📐 Step-by-Step Explanation
Diameter = 10 → radius r = 5 cm
Area = πr² = 3.14 × 5² = 3.14 × 25 = 78.5 cm²
✅ Trap A: using diameter (10) instead of radius → 3.14 × 100 = 314 (10× too large!)
✅ Trap C: forgetting to square → 3.14 × 5 only.
15
Triangle Angles
In triangle ABC, angle A = 2x + 10°, angle B = 3x − 5°, and angle C = x + 15°. Find the measure of the largest angle.
KEY: SUM of interior angles of any triangle = 180°
📐 Step-by-Step Explanation
(2x+10) + (3x−5) + (x+15) = 180
6x + 20 = 180 → 6x = 160 → x = 160/6 ≈ 26.67°
Angle A = 2(26.67)+10 ≈ 63.3°
Angle B = 3(26.67)−5 ≈ 75.0° ← largest
Angle C = 26.67+15 ≈ 41.7°
Check: 63.3+75+41.7 = 180 ✓
✅ The largest angle is Angle B ≈ 75°. Always find all angles and compare!
16
Volume · Rectangular Prism
A rectangular box is 8 cm long, 5 cm wide, and 3 cm tall. If the height is doubled, what happens to the volume?
Volume of Rectangular Prism = length × width × height
KEY: Volume is DIRECTLY proportional to each dimension → double one = double volume
📐 Step-by-Step Explanation
Original V = 8 × 5 × 3 = 120 cm³
New height = 6 cm → V = 8 × 5 × 6 = 240 cm³
240 is exactly twice 120 ✓
✅ Key insight: doubling one dimension doubles the volume. Only doubling ALL three would × 8.
17
Parallel Lines & Transversals
Two parallel lines are cut by a transversal. One angle measures 65°. Which of the following could be the measure of a co-interior (same-side interior) angle?
Co-interior (also called "consecutive interior" or "same-side interior") angles are supplementary.
65° + co-interior = 180° → co-interior = 115°
✅ Alternate interior angles would be 65° (equal). Trap: 65° is for alternate interior, not co-interior!
18
Similar Triangles
Triangle ABC is similar to triangle DEF. The sides of ABC are 6, 8, and 10. The shortest side of DEF is 9. What is the longest side of DEF?
KEY: SIMILAR → all sides in the SAME RATIO · find scale factor first!
📐 Step-by-Step Explanation
Shortest side of ABC = 6, shortest side of DEF = 9
Scale factor = 9/6 = 3/2 (or 1.5)
Longest side of ABC = 10
Longest side of DEF = 10 × (3/2) = 15
Check: middle side = 8 × 1.5 = 12 → DEF has sides 9, 12, 15 ✓
19
Coordinate Geometry · Distance
What is the distance between points A(1, 2) and B(5, −1)? Leave your answer in simplified radical form.
Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]
KEY: Distance = PYTHAGOREAN THEOREM on the coordinate plane → (Δx)² + (Δy)²
📐 Step-by-Step Explanation
Δx = 5 − 1 = 4, Δy = −1 − 2 = −3
Distance = √(4² + (−3)²) = √(16 + 9) = √25 = 5
✅ This is the 3-4-5 right triangle again! (−3)² = +9, not −9.
20
Composite Figures · Area
A figure consists of a rectangle (length 10, width 6) with a semicircle attached to one of its shorter sides (diameter = 6). What is the total area? (Use π ≈ 3.14)
KEY: COMPOSITE AREA = break into PARTS → add each area separately
📐 Step-by-Step Explanation
Rectangle area = 10 × 6 = 60
Semicircle: diameter = 6 → radius = 3
Semicircle area = ½πr² = ½ × 3.14 × 9 = 4.5π ≈ 14.13
Total = 60 + 14.13 = ≈ 74.13
✅ Trap B: using full circle instead of half. Trap A: using r=3 but forgetting the ½.