Subtract constant b from both sides, then divide by variable coefficient a.

Multiplying or dividing both sides by a negative number ALWAYS reverses the inequality symbol.

m = Rate per unit (variable change), b = Flat fee / starting amount (constant baseline at x = 0).

Amount that y increases or decreases for every 1-unit increase in x.

Baseline, initial value, or starting measurement before any change occurs.

Rise over run between any two points on the line.

Instantly determine slope and intercepts without converting to y = mx + b by hand.

Lines have identical slopes and different y-intercepts.

Slopes multiply to -1 (e.g., -3/4 becomes +4/3).

Lines have different slopes and intersect at exactly 1 coordinate point (x, y).

Lines have identical slopes but different y-intercepts (parallel lines that never meet).

Lines have identical slopes and identical y-intercepts (the same exact line).

Represents total headcount, tickets, coins, or units.

Weights each variable by its monetary value, weight, or percentage.

Must meet or exceed the threshold value.

Must stay under or equal to the ceiling value.

Strict inequality where equality is not permitted.

Dashed line means points on line are NOT solutions; solid line means points on line ARE solutions.

Shade the half-plane that satisfies the inequality. Origin test (0, 0) verifies correct region.

Set cross-products equal to solve for the missing variable.

Divide individual part by the total sum of all parts.

Place target units on the opposite side (top vs bottom) so unwanted units cancel out cleanly.

When converting square units, square the conversion ratio; for cubic units, cube the ratio!

Direct multiplication of decimal percent by whole base.

Always divide the difference by the ORIGINAL (Old) baseline value.

Increase by 15% ──► multiply by 1.15. Decrease by 20% ──► multiply by 0.80.

Arithmetic average of all data points.

If n is even, average the two middle numbers.

An extreme high value drags the mean up, while the median barely changes.

Higher dispersion/spread ──► Larger σ. Tightly clustered data ──► Smaller σ.

Multiply each value by its count, sum them up, and divide by total counts.

m = predicted rate of change, b = predicted initial value at x = 0.

Vertical distance from the data point to the trend line (positive if point is above line, negative if below).

Changes by a constant fixed amount per unit of time (e.g., adds 5 items each year).

Multiplies by a constant ratio or percentage per unit of time (e.g., grows by 5% each year).

Ratio of target outcomes to total possible outcomes in sample space.

Crucial: Restrict the denominator ONLY to the group specified after "given that" or "of the".

Multiply sample rate by the entire population size.

The true population parameter is expected to fall between (Estimate - MOE) and (Estimate + MOE).

Increasing sample size reduces uncertainty and narrows the margin of error.

Allows generalizing findings to the broader population from which subjects were sampled.

Proves a cause-and-effect relationship. Observational studies prove correlation only!

Two perfect squares subtracted factor into conjugate binomials.

Square of first, twice the product of both, square of last.

Find two numbers that multiply to a·c and add to b.

Add exponents when multiplying; subtract when dividing.

Multiply exponents when raising a power to another power.

Invert the base into the denominator.

Numerator = power, Denominator = root index.

Any non-zero base to the 0 power equals 1.

Only terms with the exact same variable and exponent can be combined.

Distribute every term in the first polynomial across every term in the second.

Only common FACTORS can be cancelled, never individual terms!

Quotient plus Remainder over Divisor.

Plug in x = a directly to find the remainder without performing long division.

Substitute the input value into every occurrence of the variable.

Evaluate inside function g(x) first, then plug that output into outside function f.

Clear denominators first, group all terms containing the target variable on one side, factor it out, and divide.

Solves any quadratic equation in standard form ax² + bx + c = 0.

D > 0 ──► 2 Distinct Real Solutions | D = 0 ──► 1 Real Solution | D < 0 ──► 0 Real Solutions (2 Complex)

Substitute line equation for y in the parabola equation to create a single quadratic.

A tangent line touches the parabola at exactly ONE point.

Must split into two separate linear equations.

When solving √(f(x)) = g(x), any solution making g(x) negative is extraneous.

h₀ = initial launch height, v₀ = initial vertical velocity. Time to max height = -v₀ / (2 · (-16)).

P₀ = initial value at t = 0, b = growth factor (1 + r) or decay factor (1 - r).

Displays the y-intercept as an explicit constant.

Displays the maximum or minimum value (k) as an explicit constant.

Displays the roots / zeros as explicit constants.

a = vertical stretch, b = growth/decay base, c = vertical shift.

The line the curve approaches as x approaches negative infinity (growth) or positive infinity (decay).

Plug in x = 0: y = a · b⁰ + c = a(1) + c = a + c.

Changes sign across the x-axis.

Does not change sign; vertex is tangent to the x-axis.

Values of x for which the rational function is undefined.

Circular base area (πr²) multiplied by height h.

Exactly one-third the volume of a cylinder with identical dimensions.

Four-thirds times pi times radius cubed.

B represents the area of the base polygon.

When every linear dimension is multiplied by k, surface area scales by k² and volume scales by k³.

The interior angles of any triangle always sum to 180°.

Corresponding angles are EQUAL; corresponding side lengths are strictly PROPORTIONAL.

Parallel lines cut by a transversal create pairs of equal alternate angles.

O = Opposite, A = Adjacent, H = Hypotenuse.

Multiples also form valid triples: 6-8-10, 9-12-15, 10-24-26, 30-40-50.

Short leg = x (opposite 30°), long leg = x√3 (opposite 60°), hypotenuse = 2x.

The sine of an acute angle equals the cosine of its complementary angle.

Central angle over 360° equals arc length s over circumference (2πr) equals sector area A over total area (πr²).

Applies exclusively when central angle θ is in radians.

Forms a right angle with the radius at the point of contact.

Any angle inscribed inside a semicircle is always a 90° right angle.

Multiply by π/180 to get radians; multiply by 180/π to get degrees.

On the unit circle, the x-coordinate is cos θ and the y-coordinate is sin θ.

Key angles to memorize for instant recall.

Center is (h, k) [signs flip!], Radius = √r².

Take half the middle coefficient and square it to create perfect square trinomials.