Algebra 1
0 / 10 answered
Quick Memory: Solve for x →
ISOLATE the variable. Move numbers to one side, keep x on the other. Use inverse operations (opposite of +, −, ×, ÷).📖 Worked Example
Sarah has $40. She buys notebooks at $3 each. She has $22 left.
→ Equation: 40 − 3n = 22 → 3n = 18 → n = 6 notebooks
→ Equation: 40 − 3n = 22 → 3n = 18 → n = 6 notebooks
Marcus earns $12 per hour working at a café. After working some hours, he has $156 total. He already had $12 saved before starting work. How many hours did he work?
💡 Explanation
Set up the equation:
Subtract 12 from both sides:
Divide by 12:
✓ Check:
12h + 12 = 156Subtract 12 from both sides:
12h = 144Divide by 12:
h = 12✓ Check:
12(12) + 12 = 144 + 12 = 156 ✔
Quick Memory: When you
MULTIPLY or DIVIDE both sides by a negative number, FLIP THE INEQUALITY SIGN! (e.g., < becomes >)📖 Worked Example
Solve −2x < 8
Divide by −2 (flip sign!): x > −4
Divide by −2 (flip sign!): x > −4
Solve the inequality: −3x + 7 ≥ 19
⚠️ Watch out — a negative coefficient is hiding here!
⚠️ Watch out — a negative coefficient is hiding here!
💡 Explanation
−3x + 7 ≥ 19Subtract 7:
−3x ≥ 12Divide by −3 → FLIP THE SIGN!:
x ≤ −4Common mistake: forgetting to flip gives x ≥ −4 (wrong!)
Quick Memory:
SUBSTITUTION = solve one equation for a variable, then plug it into the other. Like replacing a person's name with their nickname!📖 Worked Example
Solve: y = 2x + 1 and y = −x + 7
Substitute: 2x + 1 = −x + 7 → 3x = 6 → x = 2, y = 5
Substitute: 2x + 1 = −x + 7 → 3x = 6 → x = 2, y = 5
Two friends are texting. Emma sends 3 more texts than Jake. Together they send 47 texts. How many texts does Jake send?
💡 Explanation
Let Jake =
Jake sends 22 texts, Emma sends 25 texts. Total: 47 ✔
j, Emma = j + 3j + (j + 3) = 47 → 2j + 3 = 47 → 2j = 44 → j = 22Jake sends 22 texts, Emma sends 25 texts. Total: 47 ✔
Quick Memory:
SLOPE = RISE ÷ RUN = (y₂ − y₁) ÷ (x₂ − x₁). Think of it as: How steep is the hill?📖 Worked Example
Find slope through (1, 3) and (4, 9):
m = (9 − 3) ÷ (4 − 1) = 6 ÷ 3 = 2
m = (9 − 3) ÷ (4 − 1) = 6 ÷ 3 = 2
A line passes through the points (−2, 5) and (4, −7). What is the slope of the line?
💡 Explanation
m = (y₂ − y₁) / (x₂ − x₁) = (−7 − 5) / (4 − (−2))= −12 / 6 = −2Watch the double negatives:
4 − (−2) = 4 + 2 = 6
Quick Memory: Exponent rules:
x⁰ = 1 (anything to zero = 1), xᵃ · xᵇ = xᵃ⁺ᵇ (same base → ADD exponents), x⁻ⁿ = 1/xⁿ (negative → FLIP to denominator)📖 Worked Example
Simplify 2³ × 2⁴ = 2⁷ = 128
Which expression is equal to 5⁰ + 3⁻¹?
⚠️ Tricky: both rules apply here at the same time!
⚠️ Tricky: both rules apply here at the same time!
💡 Explanation
5⁰ = 1 (any nonzero number to the 0 power = 1)3⁻¹ = 1/3 (negative exponent = reciprocal)1 + 1/3 = 3/3 + 1/3 = 4/3
Quick Memory: To factor
x² + bx + c, find two numbers that MULTIPLY to c and ADD to b. Chant: "Times C, Plus B!"📖 Worked Example
Factor x² + 5x + 6
Need: × = 6, + = 5 → use 2 and 3 → (x + 2)(x + 3)
Need: × = 6, + = 5 → use 2 and 3 → (x + 2)(x + 3)
Factor the quadratic: x² − 7x + 12
Hint: both numbers you need will be negative — why?
Hint: both numbers you need will be negative — why?
💡 Explanation
Find two numbers: multiply to
→
Answer:
Verify: FOIL →
+12, add to −7→
−3 × −4 = 12 ✓ and −3 + (−4) = −7 ✓Answer:
(x − 3)(x − 4)Verify: FOIL →
x² − 4x − 3x + 12 = x² − 7x + 12 ✔
Quick Memory: A
FUNCTION maps each input (x) to exactly one output (y). Check: VERTICAL LINE TEST — if a vertical line hits the graph twice, it's NOT a function!📖 Worked Example
Given f(x) = 3x − 2, find f(4):
f(4) = 3(4) − 2 = 12 − 2 = 10
f(4) = 3(4) − 2 = 12 − 2 = 10
Given f(x) = x² − 2x + 5, what is f(−3)?
💡 Explanation
Substitute
Watch:
x = −3:f(−3) = (−3)² − 2(−3) + 5= 9 − (−6) + 5 = 9 + 6 + 5 = 20Watch:
−2(−3) = +6, not −6!
Quick Memory: Percent Change =
(NEW − OLD) ÷ OLD × 100%. Remember: OLD is always the denominator!📖 Worked Example
Price goes from $50 to $65. Percent increase?
(65 − 50) / 50 × 100 = 30%
(65 − 50) / 50 × 100 = 30%
A jacket was originally priced at $80. After a discount, it costs $56. What is the percent decrease?
💡 Explanation
Percent decrease = (80 − 56) / 80 × 100= 24 / 80 × 100 = 0.30 × 100 = 30%Common error: dividing by the NEW price (56) gives 42.9% — always divide by the ORIGINAL!
Quick Memory:
|x| = k means two solutions: x = k OR x = −k. Think: distance from zero can go left OR right!📖 Worked Example
Solve |x − 2| = 5:
Case 1: x − 2 = 5 → x = 7
Case 2: x − 2 = −5 → x = −3
Case 1: x − 2 = 5 → x = 7
Case 2: x − 2 = −5 → x = −3
Solve for all values of x: |2x − 3| = 11
💡 Explanation
Split into two cases:
Case 1:
Case 2:
Answer:
Case 1:
2x − 3 = 11 → 2x = 14 → x = 7Case 2:
2x − 3 = −11 → 2x = −8 → x = −4Answer:
x = 7 or x = −4
Quick Memory: For mixture/rate problems: set up a
TABLE with columns: Amount × Rate = Total. Each row is one part of the mixture.📖 Worked Example
Mix 10 kg of $3/kg coffee with 5 kg of $6/kg coffee.
Total cost: 10(3) + 5(6) = 30 + 30 = $60 for 15 kg → $4/kg
Total cost: 10(3) + 5(6) = 30 + 30 = $60 for 15 kg → $4/kg
A chemist mixes a 20% acid solution with a 50% acid solution to get 60 mL of a 30% acid solution. How many mL of the 20% solution does she use?
💡 Explanation
Let x = mL of 20% solution, then (60 − x) = mL of 50% solution
0.20x + 0.50(60 − x) = 0.30(60)0.20x + 30 − 0.50x = 18−0.30x = −12 → x = 40 mL
Geometry
0 / 10 answered
Quick Memory: Parallel lines cut by a transversal:
ALTERNATE INTERIOR angles are equal (Z-shape). CO-INTERIOR (same side) angles add to 180° (C-shape).📖 Worked Example
Two parallel lines, transversal crosses them. One angle = 65°.
Alternate interior angle = 65° (equal). Co-interior angle = 180° − 65° = 115°.
Alternate interior angle = 65° (equal). Co-interior angle = 180° − 65° = 115°.
Two parallel lines are cut by a transversal. One angle formed measures 3x + 15° and its co-interior angle (same-side interior) measures 2x + 25°. Find x.
💡 Explanation
Co-interior angles are supplementary (add to 180°):
(3x + 15) + (2x + 25) = 1805x + 40 = 180 → 5x = 140 → x = 28
Quick Memory:
a² + b² = c² where c is always the hypotenuse (longest side, opposite the right angle). Common triples: 3-4-5, 5-12-13, 8-15-17.📖 Worked Example
Legs = 6 and 8. Find hypotenuse:
c² = 6² + 8² = 36 + 64 = 100 → c = 10
c² = 6² + 8² = 36 + 64 = 100 → c = 10
A 17-foot ladder leans against a wall. The base of the ladder is 8 feet from the wall. How high up the wall does the ladder reach?
💡 Explanation
Ladder = hypotenuse (17 ft), base = one leg (8 ft), height = other leg
This is the 8-15-17 Pythagorean triple!
8² + h² = 17²64 + h² = 289 → h² = 225 → h = 15 ftThis is the 8-15-17 Pythagorean triple!
Quick Memory: Composite shapes →
SPLIT into simple shapes (rectangles, triangles, circles). Find each area separately, then ADD or SUBTRACT.📖 Worked Example
L-shaped figure: big rectangle 8×6 minus small rectangle 3×2
Area = 48 − 6 = 42 sq units
Area = 48 − 6 = 42 sq units
A rectangular yard is 12 m × 10 m. Inside it, there is a circular flower bed with a diameter of 4 m. What is the area of the yard outside the flower bed? (Use π ≈ 3.14)
💡 Explanation
Rectangle area:
Circle radius = 4/2 = 2 m
Circle area:
Outside area:
12 × 10 = 120 m²Circle radius = 4/2 = 2 m
Circle area:
π × r² = 3.14 × 4 = 12.56 m²Outside area:
120 − 12.56 = 107.44 m²
Quick Memory: Triangle Angle Sum =
180° always! Exterior angle = sum of the two NON-adjacent interior angles. (Exterior angle theorem)📖 Worked Example
Triangle angles: 50°, 70°, and ?
? = 180 − 50 − 70 = 60°
? = 180 − 50 − 70 = 60°
In a triangle, one exterior angle measures 130°. The two non-adjacent interior angles are in the ratio 3 : 2. What are the two interior angles?
💡 Explanation
Exterior angle = sum of two remote interior angles:
Angles:
Check:
3x + 2x = 1305x = 130 → x = 26Angles:
3(26) = 78° and 2(26) = 52°Check:
78 + 52 = 130° ✔
Quick Memory:
Cylinder V = πr²h. Cone V = (1/3)πr²h. Pyramid V = (1/3) × Base × h. The 1/3 rule: cones and pyramids hold 1/3 of their "full" version!📖 Worked Example
Cylinder: r = 3, h = 5 → V = π(9)(5) = 45π ≈ 141.3
A cone and a cylinder have the same radius (r = 4 cm) and the same height (h = 9 cm). What is the ratio of the volume of the cone to the cylinder?
💡 Explanation
V_cone = (1/3)πr²h and V_cylinder = πr²hSince they share the same r and h:
V_cone / V_cylinder = (1/3)πr²h / πr²h = 1/3Ratio =
1 : 3
Quick Memory: Similar triangles have
EQUAL ANGLES and PROPORTIONAL SIDES. Set up: small/large = small/large. Cross multiply to solve!📖 Worked Example
Triangles similar. Small: sides 3, 4, 5. Large: one side = 12.
Scale factor = 12/4 = 3 → other sides: 9 and 15
Scale factor = 12/4 = 3 → other sides: 9 and 15
Two similar triangles: the smaller triangle has sides 6, 8, and 10. The largest side of the bigger triangle is 25. What is the perimeter of the larger triangle?
💡 Explanation
Scale factor = 25 / 10 = 2.5
Larger triangle sides:
Perimeter =
Larger triangle sides:
6 × 2.5 = 15, 8 × 2.5 = 20, 10 × 2.5 = 25Perimeter =
15 + 20 + 25 = 60
Quick Memory: Arc length =
(θ/360) × 2πr. Sector area = (θ/360) × πr². The fraction θ/360 = "what fraction of the whole circle?" — apply it to circumference or area!📖 Worked Example
Radius = 6, central angle = 90°
Arc = (90/360) × 2π(6) = (1/4)(12π) = 3π ≈ 9.42
Arc = (90/360) × 2π(6) = (1/4)(12π) = 3π ≈ 9.42
A circle has a radius of 10 cm. A sector has a central angle of 72°. What is the area of the sector? (Use π ≈ 3.14)
💡 Explanation
A_sector = (θ/360) × πr²= (72/360) × 3.14 × 10²= (1/5) × 3.14 × 100= (1/5) × 314 = 62.8 cm²
Quick Memory:
MIDPOINT = ((x₁+x₂)/2 , (y₁+y₂)/2). DISTANCE = √((x₂−x₁)² + (y₂−y₁)²). Midpoint = average the coordinates!📖 Worked Example
Midpoint of (2, 4) and (8, 10):
M = ((2+8)/2, (4+10)/2) = (5, 7)
M = ((2+8)/2, (4+10)/2) = (5, 7)
The midpoint of segment AB is M(3, −1). If A = (−1, 5), what are the coordinates of point B?
💡 Explanation
Midpoint formula:
For x:
For y:
M = ((x_A + x_B)/2, (y_A + y_B)/2)For x:
(−1 + x_B)/2 = 3 → −1 + x_B = 6 → x_B = 7For y:
(5 + y_B)/2 = −1 → 5 + y_B = −2 → y_B = −7B = (7, −7)
Quick Memory: Surface area of a rectangular prism:
SA = 2(lw + lh + wh). Think of it as: 6 rectangles — top+bottom, front+back, left+right — each pair × 2!📖 Worked Example
Box: l=3, w=4, h=5 → SA = 2(12 + 15 + 20) = 2(47) = 94
A gift box is shaped like a rectangular prism with dimensions 8 cm × 5 cm × 3 cm. What is the minimum amount of wrapping paper needed to cover it completely?
💡 Explanation
SA = 2(lw + lh + wh)= 2(8×5 + 8×3 + 5×3)= 2(40 + 24 + 15)= 2(79) = 158 cm²
Quick Memory: Reflections:
over x-axis → (x, −y). Over y-axis → (−x, y). Over y = x → (y, x). Just SWAP or NEGATE the right coordinate!📖 Worked Example
Point (3, −5) reflected over the y-axis:
Negate x → (−3, −5)
Negate x → (−3, −5)
Point P(−4, 7) is first reflected over the x-axis, then translated by the vector (+3, −2). What are the final coordinates?
💡 Explanation
Step 1 — Reflect over x-axis:
Step 2 — Translate by (+3, −2):
(x, y) → (x, −y)P(−4, 7) → P'(−4, −7)Step 2 — Translate by (+3, −2):
P'(−4 + 3, −7 + (−2)) = P''(−1, −9)