Part 1 — Linear Functions, Equations & Inequalities
⚡
Quick Memory Points
Slope-Intercept: y = mx + b | m = slope (rise/run), b = y-intercept Solving: Isolate the variable — do the same to both sides Inequality flip: Multiply or divide by negative → flip the sign (< becomes >) Two special lines: Horizontal → slope = 0 | Vertical → slope = undefined
1
Algebra★☆☆ Easy
What is the slope of the line y = 3x − 5?
Look at the coefficient in front of x in slope-intercept form.
Explanation
In slope-intercept form y = mx + b, the slope is m — the number multiplied by x. Here, y = 3x − 5, so m = 3. The −5 is the y-intercept, not the slope. Common mistake: confusing slope (m) with y-intercept (b).
2
Algebra★☆☆ Easy
Solve for x: 2x + 7 = 15
Explanation
Step 1: Subtract 7 from both sides → 2x = 8
Step 2: Divide both sides by 2 → x = 4
Always undo addition/subtraction first, then multiplication/division.
3
Algebra★★☆ Medium
Which graph represents a line with slope −2 and y-intercept 4?
Slope is negative → line goes down from left to right. Y-intercept = where it crosses the y-axis.
Explanation
y = −2x + 4: negative slope means the line falls (goes down) left to right. The y-intercept is +4, so it crosses the y-axis at (0, 4). Option B has a positive slope (rises). Option C has the wrong intercept. Option D ignores slope entirely.
4
Algebra★★☆ Medium
Solve the inequality: −3x > 12
⚠ Dividing by a negative number FLIPS the inequality sign!
Explanation
Divide both sides by −3. Because you divide by a negative, flip > to <: x < 12 ÷ (−3) = −4
So x < −4. This is the most commonly missed concept in inequalities — always flip when multiplying or dividing by a negative!
5
Algebra★★☆ Medium
What is the slope of a line passing through (2, 5) and (6, 13)?
slope = (y₂ − y₁) / (x₂ − x₁)
Explanation
Using the slope formula: m = (13 − 5) / (6 − 2) = 8 / 4 = 2
Common error: mixing up Δy and Δx order, or putting x values in the numerator. Remember: rise (y-change) over run (x-change).
6
Algebra★★★ Tricky
Which value of x satisfies BOTH inequalities? x > −1 AND x ≤ 3
Both conditions must be true at the same time. Think: AND = overlap region.
Explanation
We need −1 < x ≤ 3 (both conditions). Check each:
A: −2 > −1? ✗ No.
B: −1 > −1? ✗ Strict inequality, −1 is not allowed.
C: 0 > −1? ✓ 0 ≤ 3? ✓ Both pass!
D: 4 ≤ 3? ✗ No.
7
Algebra★★★ Tricky
What is the y-intercept of the line passing through (3, 7) with slope 2?
y − y₁ = m(x − x₁) → then solve for y = mx + b
Explanation
Use point-slope: y − 7 = 2(x − 3)
→ y = 2x − 6 + 7 = 2x + 1
So b = 1. Trap: many students forget to distribute the slope into both the x and the constant.
8
Algebra★★★ Tricky
Two lines are parallel. Line 1 has equation y = 4x + 1. What could be the equation of Line 2?
Parallel lines = same slope, different y-intercept. Perpendicular = slopes multiply to −1.
Explanation
Parallel lines have the same slope but different intercepts. Line 1's slope = 4. Option C has slope 4 but intercept −7 — that's parallel! Option B is the same line (not parallel, identical). Option A has slope −1/4 (perpendicular). Option D has slope 1/4 (neither).
9
Algebra★★☆ Medium
A linear function has the equation f(x) = −x + 6. What is f(−2)?
Explanation
Substitute x = −2: f(−2) = −(−2) + 6 = 2 + 6 = 8
Common error: −(−2) = −2 (wrong). Negative × negative = positive. So −(−2) = +2.
10
Algebra★★★ Tricky
Which inequality is graphed by a dashed line with shading ABOVE it?
Dashed = strict inequality (< or >). Above = y is greater than the line.
Explanation
Dashed line → strict inequality (no equal sign, so not ≤ or ≥). Shading above → y is greater than the line, so >. Therefore: y > 2x + 1.
Memory trick: solid = includes the line (≤ or ≥) · dashed = excludes the line (< or >)
Part 2 — Triangles, Polygons & Circles
📐
Quick Memory Points
Triangle angles: Always sum to 180° Pythagorean theorem: a² + b² = c² (right triangle, c = hypotenuse) Polygon interior angles: Sum = (n − 2) × 180° Circle: C = 2πr · A = πr² · Arc = (θ/360) × 2πr
11
Geometry★☆☆ Easy
A triangle has angles 55° and 80°. What is the third angle?
Explanation
All triangle angles add to 180°: 55 + 80 + ? = 180 ? = 180 − 135 = 45°
12
Geometry★★☆ Medium
A right triangle has legs of length 6 and 8. What is the length of the hypotenuse?
a² + b² = c²
Explanation
6² + 8² = 36 + 64 = 100 c = √100 = 10
This is the classic 3-4-5 Pythagorean triple scaled by 2: (6, 8, 10). Memorizing common triples saves time: 3-4-5, 5-12-13, 8-15-17.
13
Geometry★★☆ Medium
What is the sum of interior angles of a hexagon (6 sides)?
Sum = (n − 2) × 180°
Explanation
(6 − 2) × 180 = 4 × 180 = 720°
Quick reference: triangle=180, quadrilateral=360, pentagon=540, hexagon=720. Each extra side adds 180°.
14
Geometry★★☆ Medium
A circle has radius 5. What is its circumference? (Use π ≈ 3.14)
C = 2πr
Explanation
C = 2 × 3.14 × 5 = 31.4
Option B (78.5) = area (πr²), not circumference. Common mix-up: C = 2πr vs A = πr². Remember: Circumference = around (like a perimeter), Area = inside.
15
Geometry★★★ Tricky
Triangle ABC is isosceles with AB = AC. If angle A = 40°, what are angles B and C?
Isosceles = two equal sides → the two BASE angles are equal.
Explanation
In isosceles triangle, base angles (B and C) are equal. Remaining angle sum: 180 − 40 = 140°. Split equally: 140 ÷ 2 = 70°. So B = C = 70°.
16
Geometry★★★ Tricky
An exterior angle of a triangle equals which of the following?
Exterior Angle Theorem — a super useful shortcut!
Explanation
Exterior Angle Theorem: An exterior angle = sum of the two remote interior angles (the ones not touching it). Example: if a triangle has angles 50°, 70°, 60°, the exterior angle at the 60° vertex = 50 + 70 = 120°. Quick check: 120 + 60 = 180° ✓ (straight line).
17
Geometry★★☆ Medium
What is the area of a circle with diameter 10? (Use π ≈ 3.14)
Diameter = 2 × radius. Don't forget to halve it before using A = πr²!
Explanation
Diameter = 10, so radius r = 5. A = π × 5² = 3.14 × 25 = 78.5
Trap: Option A uses diameter (10) instead of radius. Always convert diameter → radius first!
18
Geometry★★★ Tricky
Each interior angle of a regular polygon measures 135°. How many sides does it have?
Two triangles are similar. The sides of the smaller triangle are 3, 4, 5. If the longest side of the larger triangle is 15, what is its shortest side?
Similar triangles = same shape, proportional sides. Find the scale factor first.
Explanation
Scale factor: largest side 5 → 15, so scale = 15/5 = 3. Multiply all sides by 3: 3×3=9, 4×3=12, 5×3=15
Shortest side of larger triangle = 9.
20
Geometry★★★ Tricky
A central angle of 90° cuts a circle with radius 6. What is the arc length?
Arc length = (θ / 360) × 2πr
Explanation
Arc = (90/360) × 2π × 6 = (1/4) × 12π = 3π ≈ 9.42
90° is exactly 1/4 of the full circle (360°), so arc = 1/4 of circumference. Full circumference = 2π×6 = 12π → 12π/4 = 3π.