To isolate the variable in a linear inequality, solve as you would an equation, but reverse the sign when multiplying or dividing both sides by a negative.

a x + b \leq c

\Rightarrow x \leq \frac{c - b}{a}

if a > 0;

x \geq \frac{c - b}{a}

if a < 0

Lesson: Linear inequalities

Compound inequalities

Every algebraic step in a three-part inequality goes to all three parts, with both signs reversing when you multiply or divide by a negative.

a < b x + c < d

Lesson: Linear inequalities

Range of a product of two bounded variables

min (a c, a d, b c, b d) \leq x y \leq max (a c, a d, b c, b d)

when

a \leq x \leq b,

c \leq y \leq d

Lesson: Linear inequalities

Quadratics

10 formulas

Course: Quadratic Equations Practice: Algebra

Sum and product of the roots of a quadratic

r + s = - \frac{b}{a}

and

r s = \frac{c}{a}

Lesson: Solving quadratics and roots

Standard form for factoring a quadratic

a x^{2} + b x + c = 0

Lesson: Solving quadratics and roots

The difference of two squares factors into the product of the sum and difference of their square roots.

x^{2} - k^{2} = (x - k) (x + k)

Lesson: Solving quadratics and roots

The roots of ax^2 + bx + c = 0

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Lesson: Solving quadratics and roots

The sum and product of the roots of ax^2 + bx + c = 0

r + s = - \frac{b}{a},

r s = \frac{c}{a}

Lesson: Solving quadratics and roots

The sum of the squares of the roots, in terms of the sum and product

r^{2} + s^{2} = (r + s)^{2} - 2 r s

Lesson: Solving quadratics and roots

Clearing a denominator from a relation with a variable in the denominator

x + \frac{a}{x} = b

⟹ x^{2} - b x + a = 0

Lesson: Quadratics in context

Product of two quantities with a known difference $k$

n (n + k) = P

Lesson: Quadratics in context

Total revenue is the product of price and quantity.

R = p \times q

Lesson: Quadratics in context

Discriminant

Δ = b^{2} - 4 a c

Δ > 0 : two real roots

Δ = 0 : one real root

Δ < 0 : no real roots

Functions and sequences

8 formulas

Course: Functions and Sequences Practice: Algebra

Evaluating a function at a given input

f (a) means replace every x with a

Lesson: Functions and custom operations

Evaluating a two-input custom operation

x # y = R (x, y)

Lesson: Functions and custom operations

A recursive rule builds each term from the one before it.

a_{n + 1} = a_{n} + d

or

a_{n + 1} = r \cdot a_{n}

Lesson: Sequences

The nth term of an arithmetic sequence

a_{n} = a_{1} + (n - 1) d

Lesson: Sequences

The common difference from any two terms

d = \frac{a _{j} - a _{i}}{j - i}

Lesson: Sequences

The nth term of a geometric sequence from its first term and common ratio

a_{n} = a_{1} r^{n - 1}

Lesson: Sequences

Position of a given value in an arithmetic sequence

n = 1 + \frac{a _{n} - a _{1}}{d}

Lesson: Sequences

The sum of an arithmetic sequence equals the number of terms times the average of the first and last terms.

S_{n} = \frac{n ( a _{1} + a _{n} )}{2}

Lesson: Sequences

Percents and interest

13 formulas

Course: Percents and Interest Practice: Arithmetic

The part equals the decimal form of the percent multiplied by the whole.

part = \frac{percent}{100} \times whole

Lesson: Percent of a whole

Percent change, as a percent of the original value

percent change = \frac{new - original}{original} \times 100

Lesson: Percent change

Recover the original by dividing the new value by the multiplier

O = \frac{N}{m}

Lesson: Percent change

List price from cost with a markup percent, and selling price from list price with a discount percent

L = C (1 + \frac{m}{100})

and

S = L (1 - \frac{d}{100})

Lesson: Percent change

Net multiplier for two successive changes

M_{net} = (1 + \frac{p _{1}}{100}) (1 + \frac{p _{2}}{100})

Lesson: Percent change

A quantity A that is p percent more than a base B

A = B (1 + \frac{p}{100})

Lesson: Percent comparisons

If A is p% more than B, then B is q% less than A, where

q = \frac{100 p}{100 + p}

Lesson: Percent comparisons

The combined multiplier for a chain of two comparisons

M_{A, C} = M_{A, B} \times M_{B, C}

Lesson: Percent comparisons

When p percent of x equals q percent of y

\frac{p}{100} \cdot x = \frac{q}{100} \cdot y

Lesson: Percent comparisons

Simple interest adds a fixed amount each period, while compound interest multiplies the balance each period.

I = P r t

and

B = P (1 + \frac{r}{n})^{n t}

Lesson: Simple and compound interest

Balance from principal and interest

B = P + I

Lesson: Simple and compound interest

The gap between compound and simple balance after two years, same principal and rate

P (1 + r)^{2} - P (1 + 2 r) = P r^{2}

Lesson: Simple and compound interest

Rate from two consecutive balances

r = \frac{B _{2}}{B _{1}} - 1

Lesson: Simple and compound interest

Ratios, rates and work

18 formulas

Course: Ratios, rates, and work Practice: Arithmetic

A ratio a:b means the actual amounts are a times a multiplier and b times that multiplier.

amount_{1} : amount_{2} = a x : b x

Lesson: Ratios and proportions

Each part's fraction of the whole, from ratio a:b

\frac{a}{a + b}

and

\frac{b}{a + b}

Lesson: Ratios and proportions

When a ratio a:b gives amounts ax and bx, a known difference d between the two amounts fixes x.

(a - b) x = d

Lesson: Ratios and proportions

For a direct proportion, set the two ratios equal with matching units in matching positions.

\frac{a}{b} = \frac{c}{d}

Lesson: Ratios and proportions

For inverse proportions, the product of the two matching quantities is the same in every scenario.

x_{1} y_{1} = x_{2} y_{2}

Lesson: Ratios and proportions

The new ratio, after changes are applied to the old amounts

\frac{a x \pm p}{b x \pm q} = \frac{c}{d}

Lesson: Ratios that change

New ratio when one part changes by amount p

\frac{a x + p}{b x} = \frac{c}{d}

Lesson: Ratios that change

New ratio after both parts change by different amounts

\frac{a x + p}{b x + q} = \frac{c}{d}

Lesson: Ratios that change

A percent change to one part of the ratio becomes a concrete addition or subtraction.

\frac{a x \pm \frac{p}{100} ( a x )}{b x} = \frac{c}{d}

Lesson: Ratios that change

The distance traveled at a constant rate

d = r \cdot t

Lesson: Distance, rate, and time

Average speed for two equal legs at speeds a and b

\frac{2 ab}{a + b}

Lesson: Distance, rate, and time

A worker's rate is the amount of work completed per unit of time.

rate = \frac{work}{time}

Lesson: Work rates

Combined time for two workers with solo times a and b

T_{combined} = \frac{ab}{a + b}

Lesson: Work rates

The unknown solo rate is the combined rate minus the known solo rate.

\frac{1}{b} = \frac{1}{t} - \frac{1}{a}

Lesson: Work rates

Staggered schedules

Work done in one phase equals the combined rate times the phase time; the rows of the table sum to one full job.

W_{i} = R_{i} \times t_{i},

i \sum W_{i} = 1

Lesson: Work rates

Time to complete a remaining fraction f at rate r.

t = \frac{f}{r}

Lesson: Work rates

The weighted average is the sum of each group's average times its size, divided by the total size.

Weighted average = \frac{a _{1} n _{1} + a _{2} n _{2} + \dots + a _{k} n _{k}}{n _{1} + n _{2} + \dots + n _{k}}

Lesson: Mixtures and weighted averages

Mixing solutions

The ingredient amount in each solution is its concentration times its volume; ingredient amounts add while volumes add.

c_{1} v_{1} + c_{2} v_{2} = c (v_{1} + v_{2})

Lesson: Mixtures and weighted averages

Lines, angles and triangles

15 formulas

Course: Lines, angles and triangles Practice: Geometry

The angles on a straight line sum to 180 degrees

angle_{1} + angle_{2} + \dots = 18 0^{\circ}

Lesson: Lines and angles

The sum of all angles around a point

θ_{1} + θ_{2} + \dots + θ_{n} = 36 0^{\circ}

Lesson: Lines and angles

Sum of angle expressions on a line

a + b + c + \dots = 180

Lesson: Lines and angles

The sum of the interior angles of a triangle

A + B + C = 18 0^{\circ}

Lesson: Triangle properties

The third side of a triangle, given the other two sides

a - b < x < a + b

Lesson: Triangle properties

The area of a triangle is one half times the base times the perpendicular height.

A = \frac{1}{2} bh

Lesson: Triangle properties

The relationship between the two legs and the hypotenuse of a right triangle

a^{2} + b^{2} = c^{2}

Lesson: Right triangles

The side ratio for a 45-45-90 triangle, where x is the length of each leg and h is the hypotenuse.

legs : legs : hypotenuse = 1 : 1 : 2

h = x 2

Lesson: Right triangles

Side lengths of a 30-60-90 triangle, where x is the side opposite the 30-degree angle

short leg = x,

long leg = x 3,

hypotenuse = 2 x

Lesson: Right triangles

If two triangles have equal corresponding angles, their corresponding sides are proportional.

\frac{a}{a ^{'}} = \frac{b}{b ^{'}} = \frac{c}{c ^{'}}

Lesson: Similar triangles

For similar triangles, the ratio of areas equals the square of the ratio of corresponding sides.

\frac{A _{1}}{A _{2}} = (\frac{s _{1}}{s _{2}})^{2}

Lesson: Similar triangles

A line parallel to one side of a triangle creates a smaller similar triangle.

\frac{a}{a ^{'}} = \frac{b}{b ^{'}}

Lesson: Similar triangles

Right triangle similarity

When a right triangle is split by an altitude to the hypotenuse, the altitude is the geometric mean of the two hypotenuse segments, and each leg is the geometric mean of the full hypotenuse and its adjacent segment

h = pq,

a = p c,

b = q c

Lesson: Similar triangles

Angles on a straight line sum to 180 degrees

α + β = 180°

Lesson: Figures not drawn to scale

Valid third side of a triangle with sides a and b

∣ a - b ∣ < c < a + b

Lesson: Figures not drawn to scale

Polygons, circles and solids

18 formulas

Course: Polygons, Circles, and Solids Practice: Geometry

Total interior angles of an n-sided polygon

Sum of angles = (n - 2) \times 18 0^{\circ}

Lesson: Quadrilaterals and polygons

Area of a parallelogram

A = b \times h

Lesson: Quadrilaterals and polygons

The area of a trapezoid equals the average of the two parallel bases times the perpendicular height.

area = \frac{b _{1} + b _{2}}{2} \times h

Lesson: Quadrilaterals and polygons

The diagonal of a rectangle from its length and width

d^{2} = l^{2} + w^{2}

Lesson: Quadrilaterals and polygons

Perimeter of a polygon with sides s₁, s₂, …, sₙ

P = s_{1} + s_{2} + \dots + s_{n}

Lesson: Quadrilaterals and polygons

The circumference and area of a circle in terms of its radius

circumference = 2 π r

and

area = π r^{2}

Lesson: Circles, arcs, and sectors

The radius is half the diameter

r = \frac{d}{2}

Lesson: Circles, arcs, and sectors

Arc length as a fraction of the circumference

arc length = \frac{θ}{360} \times 2 π r

Lesson: Circles, arcs, and sectors

Area of a sector

A_{sector} = (\frac{θ}{360}) π r^{2}

Lesson: Circles, arcs, and sectors

Tangent lines

The radius to a point of tangency is perpendicular to the tangent line, forming a right triangle with the tangent segment and the distance to an external point.

r^{2} + t^{2} = d^{2}

Lesson: Circles, arcs, and sectors

Half the chord length, the distance from the center to the chord, and the radius form a right triangle.

(\frac{c}{2})^{2} + d^{2} = r^{2}

Lesson: Circles, arcs, and sectors

Area of a composite figure from its component areas

A_{composite} = \sum A_{added}

- \sum A_{removed}

Lesson: Composite figures

Right triangles in circles

A triangle inscribed in a circle with one side along the diameter has a right angle opposite that side, so the diameter is the hypotenuse.

a^{2} + b^{2} = (2 r)^{2}

Lesson: Composite figures

Volume of a rectangular solid

V = l \times w \times h

Lesson: Three-dimensional solids

Volume of a cylinder

V = π r^{2} h

Lesson: Three-dimensional solids

Surface area of a rectangular solid

S A = 2 l w + 2 l h + 2 w h

Lesson: Three-dimensional solids

Total surface area of a cylinder

S A = 2 π r^{2} + 2 π r h

Lesson: Three-dimensional solids

The space diagonal of a rectangular solid from its length, width, and height

d = l^{2} + w^{2} + h^{2}

Lesson: Three-dimensional solids

Coordinate geometry

12 formulas

Course: Coordinate Geometry Practice: Algebra

Distance between two points

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Lesson: Points, distance, and midpoint

Midpoint of a segment

M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

Lesson: Points, distance, and midpoint

The midpoint of a segment with endpoints $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$

(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

Lesson: Points, distance, and midpoint

Unknown coordinates

To recover an unknown coordinate, set the known distance equal to the distance expression, or set the known midpoint coordinates equal to the midpoint averages.

(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} = d

(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2}) = (m_{x}, m_{y})

Lesson: Points, distance, and midpoint

The slope-intercept form of a line, where m is the slope and b is the y-intercept

y = m x + b

Lesson: Lines in the plane

Slope of a line through two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Lesson: Lines in the plane

The y-intercept b given slope m and a point (x₁, y₁) on the line

b = y_{1} - m \cdot x_{1}

Lesson: Lines in the plane

The x-coordinate where lines y = m₁x + b₁ and y = m₂x + b₂ intersect

x = \frac{b _{2} - b _{1}}{m _{1} - m _{2}}

Lesson: Lines in the plane

The graph of a function pairs each input x with its output y.

y = f (x)

Lesson: Graphs of functions

The vertex of a parabola and its direction of opening

y = a (x - h)^{2} + k

Lesson: Graphs of functions

X-coordinate of each intersection point

f (x) = g (x)

Lesson: Graphs of functions

Perpendicular lines

m_{1} \cdot m_{2} = - 1

Statistics

12 formulas

Course: Descriptive Statistics Practice: Data analysis

The mean is the sum divided by the count, so the sum equals the mean times the count.

mean = \frac{sum}{count}

and

sum = mean \times count

Lesson: Mean, median, and mode

The missing value, given the target mean and all other values

x = μ \cdot n - S

Lesson: Mean, median, and mode

Mean and median of an ordered set

mean = \frac{sum}{n},

median = middle

value

of

the

ordered

data

Lesson: Mean, median, and mode

The range of a set of values

range = largest value

- smallest value

Lesson: Range and standard deviation

The interval of values within k standard deviations of the mean

mean - k \cdot SD \leq x \leq mean

+ k \cdot SD

Lesson: Range and standard deviation

Range of a set

R = x_{m a x} - x_{m i n}

Lesson: Range and standard deviation

The five-number summary of an ordered data set

Minimum,

Q1,

Median (Q2),

Q3,

Maximum

Lesson: Quartiles, percentiles, and box plots

Interquartile range, the spread of the middle half

IQR = Q_{3} - Q_{1}

Lesson: Quartiles, percentiles, and box plots

The share of data within one standard deviation of the mean in a normal distribution

μ - σ \leq x \leq μ + σ

contains about 68% of the data

Lesson: Normal distribution

One-sided percent from the within-range percent

p_{k} = \frac{P _{k}}{2}

Lesson: Normal distribution

Population standard deviation

σ = \frac{\sum ( x _{i} - μ ) ^{2}}{n}

Normal distribution

P (μ - σ < X < μ + σ) \approx 68%

P (μ - 2 σ < X < μ + 2 σ) \approx 95%

P (μ - 3 σ < X < μ + 3 σ) \approx 99.7%

Counting, probability, sets and data interpretation

18 formulas

Course: Counting, Probability and Sets Course: Data Interpretation Practice: Probability Practice: Data analysis

The number of ways to arrange r of n distinct objects in a row

\frac{n !}{( n - r )!}

Lesson: Counting principle and arrangements

Arrangements of n items where k specified items must sit together

(n - k + 1)! \times k!

Lesson: Counting principle and arrangements

Number of unordered groups of r chosen from n

C (n, r) = \frac{n !}{r ! ( n - r )!}

Lesson: Combinations and choosing then arranging

Valid committees when two specific members refuse to serve together

C (n, r) - C (n - 2, r - 2)

Lesson: Combinations and choosing then arranging

Total ways to choose from distinct groups

Total = C (n_{1}, r_{1}) \times C (n_{2}, r_{2}) \times \dots \times C (n_{k}, r_{k})

Lesson: Combinations and choosing then arranging

Selections with at least one of a required type

C (n, r) - C (m, r)

Lesson: Combinations and choosing then arranging

Ordered outcomes from choosing r from n and then arranging them

C (n, r) \times r! = P (n, r)

Lesson: Combinations and choosing then arranging

Probability of an event

P = \frac{favorable outcomes}{total outcomes}

Lesson: Probability of events

Probability that event A or event B occurs, accounting for overlap

P (A or B) = P (A)

+ P (B)

- P (A and B)

Lesson: Probability of events

Probability that all independent events occur

P (all) = P (A) \times P (B) \times \dots

Lesson: Probability of events

Total for two overlapping groups, counting the both-region once

total = A

+ B

- both

+ neither

Lesson: Overlapping sets

The average of a column is the sum of its values divided by the number of rows.

average = \frac{sum of column}{number of rows}

Lesson: Reading tables

Weighted average across table rows

\overset{x}{ˉ} = \frac{\sum ( v a l u e _{i} \times w e i g h t _{i} )}{\sum w e i g h t _{i}}

Lesson: Reading tables

Average of the data rows after any summary row is removed

\overset{x}{ˉ} = \frac{\sum x _{i}}{n}

Lesson: Reading tables

Value of one segment in a stacked bar

v = h_{top} - h_{bottom}

Lesson: Reading graphs

Amount for a category, where p is the percent share and T is that pie's total

A = p \times T

Lesson: Reading graphs

A per-row ratio from two columns

ratio_{i} = \frac{numerator _{i}}{denominator _{i}}

Lesson: Percents and change from data

Combining a value from one display with a value from another

result = V_{1} \circ V_{2}

Lesson: Combining two displays

Format rules

11 formulas

Course: Quantitative Comparison Course: Numeric Entry Course: Select One or More Practice: GRE practice test

The four fixed outcomes of every quantitative comparison

A > B,

A < B,

A = B,

or D

Lesson: The four answer choices

The relationship between Quantity A and Quantity B is determined by testing multiple allowed values.

If A > B

for all valid x,

choose

A;

if

the

sign

ever

changes,

choose

D

Lesson: Testing values

Only test values within the allowed set $S$ .

x \in S

Lesson: Testing values

The entry is the exact value the question asks for.

entry = exact computed value

Lesson: Numeric entry

Count of consecutive integers from a to b, inclusive

count = b - a + 1

Lesson: Counting the possibilities

Digit restrictions

Count of valid numbers, with one allowed set per position where $n_{i}$ is the number of digits allowed in position $i$ , and the leading position excludes zero when the number must have exactly $k$ digits.