📖 Concept
Integers include all whole numbers and their negatives: … −3, −2, −1, 0, 1, 2, 3 …
|a| = a if a ≥ 0 ; |a| = −a if a < 0
- Absolute value = distance from 0 on the number line (always ≥ 0)
- Adding integers: same signs → add and keep sign; different signs → subtract and keep sign of larger absolute value
- Multiplying/dividing: same signs → positive; different signs → negative
Example
Evaluate |−7| + (−3) × 2
= 7 + (−6) = 1
Question 01
Integers & Absolute Value
What is the value of |−15| − |−9| + (−4)?
Question 02
Integers & Absolute Value
Which expression has the greatest value?
Unit 02 · Fractions & Decimals
📖 Concept
To add/subtract fractions: find LCD (Least Common Denominator). To multiply: multiply straight across. To divide: multiply by the reciprocal.
a/b ÷ c/d = a/b × d/c = ad/bc
- LCD = smallest number divisible by all denominators
- Mixed number → improper fraction: (whole × denom + numer) / denom
- To convert fraction to decimal: divide numerator by denominator
Example
3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2 = 1.5
Question 03
Fractions & Decimals
What is 2/3 + 3/4?
Question 04
Fractions & Decimals
Which fraction is equivalent to 0.375?
Unit 03 · Ratios, Rates & Proportions
📖 Concept
A ratio compares two quantities. A proportion states two ratios are equal. Use cross-multiplication to solve.
a/b = c/d ⟹ a × d = b × c
- Unit rate: rate with denominator = 1 (e.g., 60 mph)
- Scale factor: ratio of lengths on a model vs. real life
Example
Solve: 3/5 = x/20 → 5x = 60 → x = 12
Question 05
Ratios & Proportions
A car travels 150 miles in 3 hours. At the same rate, how many miles will it travel in 5 hours?
Question 06
Ratios & Proportions
If the ratio of boys to girls in a class is 3 : 5 and there are 24 boys, how many students are in the class?
📖 Concept
Percent means "per hundred." Use the percent equation to find part, whole, or percent.
Part = Percent × Whole | % Change = (Change ÷ Original) × 100
- Percent increase: new value > original → add the % to 100%
- Percent decrease: new value < original → subtract the % from 100%
- Discount price = Original × (1 − discount%)
Example
30% of 80 = 0.30 × 80 = 24
Question 07
Percent
A jacket originally costs $80. It is on sale for 25% off. What is the sale price?
Question 08
Percent
A price increased from $50 to $65. What is the percent increase?
Unit 05 · Exponents & Order of Operations
📖 Concept
PEMDAS: Parentheses → Exponents → Multiplication & Division (left to right) → Addition & Subtraction (left to right)
aᵐ × aⁿ = aᵐ⁺ⁿ | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | (aᵐ)ⁿ = aᵐⁿ | a⁰ = 1
- Negative exponent: a⁻ⁿ = 1/aⁿ
- Scientific notation: a × 10ⁿ where 1 ≤ a < 10
Example
3 + 4² × (6 − 4) ÷ 8
= 3 + 16 × 2 ÷ 8 = 3 + 32 ÷ 8 = 3 + 4 = 7
Question 09
Exponents & PEMDAS
Evaluate: 2 + 3³ − (4 × 2 + 1)
Question 10
Exponents
Which expression is equivalent to 5³ × 5⁻¹?
Unit 06 · Algebraic Expressions
📖 Concept
An algebraic expression contains variables, constants, and operations. Simplify by combining like terms (same variable AND same exponent).
Distributive Property: a(b + c) = ab + ac
- Like terms: 3x and 5x are like; 3x and 3x² are NOT like
- Coefficient: the number multiplied by the variable
- Constant term: a term with no variable
Example
Simplify: 4x − 2(3x − 5)
= 4x − 6x + 10 = −2x + 10
Question 11
Algebraic Expressions
Simplify: 5x + 3 − 2x + 7 − x
Question 12
Algebraic Expressions
What is the value of 3x² − 2x + 5 when x = −2?
Unit 07 · One-Variable Equations
📖 Concept
To solve an equation, isolate the variable using inverse operations while keeping both sides balanced.
Inverse operations: + ↔ − × ↔ ÷
- Multi-step: distribute → combine like terms → move variable terms to one side → isolate
- Check: substitute answer back into original equation
Example
Solve: 3x − 7 = 14 → 3x = 21 → x = 7
Question 13
One-Variable Equations
Solve for x: 4(x − 3) + 2 = 18
Question 14
One-Variable Equations
If 2x + 5 = 3x − 4, what is the value of x?
📖 Concept
Solve inequalities like equations, BUT flip the inequality sign when multiplying or dividing by a negative number.
If −ax > b, then x < −b/a (sign flips!)
- Open circle (○) on number line = strict inequality (< or >)
- Closed circle (●) = non-strict (≤ or ≥)
Example
Solve: −3x < 12 → x > −4 (sign flips when ÷ by −3)
Question 15
Inequalities
Solve: −2x + 6 ≤ 12. Which describes the solution?
Unit 09 · Coordinate Plane & Functions
📖 Concept
The coordinate plane has an x-axis (horizontal) and y-axis (vertical). A function is a relationship where each input (x) has exactly one output (y).
Slope = rise/run = (y₂ − y₁) / (x₂ − x₁)
- Slope-intercept form: y = mx + b (m = slope, b = y-intercept)
- Vertical line test: if any vertical line crosses the graph more than once, it's NOT a function
- Quadrants: I (+,+) II (−,+) III (−,−) IV (+,−)
Example
Slope through (1, 2) and (3, 8): (8−2)/(3−1) = 6/2 = 3
Question 16
Coordinate Plane & Slope
What is the slope of the line passing through (−2, 1) and (4, 13)?
Question 17
Linear Functions
Which equation represents a line with slope −3 and y-intercept 4?
Unit 10 · Geometry Basics
📖 Concept
Key formulas for perimeter, area, and the Pythagorean Theorem for right triangles.
Area of triangle = ½bh | Circle: A = πr², C = 2πr | a² + b² = c²
- Pythagorean theorem: applies only to RIGHT triangles; c = hypotenuse (longest side)
- Supplementary angles: sum = 180° | Complementary: sum = 90°
- Interior angles of a triangle always sum to 180°
Example
Right triangle with legs 6 and 8: c² = 6² + 8² = 36 + 64 = 100, c = 10
Question 18
Pythagorean Theorem
A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?
Question 19
Angles & Geometry
Two angles are supplementary. One angle measures 67°. What is the measure of the other angle?
Question 20
Geometry — Area
A circle has a radius of 6 cm. Which expression gives its area? (Use π)
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Answer Key & Explanations