Equations

Equation

A statement that two expressions are equal. Example: 3(x−1) + x = −x + 7

Addition Property of Equality

If x = y, then x + z = y + z. Like quantities can be added to both sides of an equation without changing the equality.

Multiplication Property of Equality

If x = y, then xz = yz. Like quantities can be multiplied to both sides of an equation without changing the equality.

Literal Equation

An equation that contains two or more variables. Example: 2x + 3y = 15

Formula

An equation that models a known relationship between two or more quantities. Formulas are a type of literal equation, but not every literal equation is a formula.

Solving for a Specified Variable

Rearranging a literal equation or formula to isolate one particular variable on one side. Example: From A = P + PRT, solving for P gives P = A / (1 + RT)

Solution Set

The set of all numbers that satisfy a given equation or inequality.

Interval Notation

A symbolic way of indicating a selected interval of the real number line using brackets [ ] and parentheses ( ).

Closed Bracket [ or ]

Used in interval notation to indicate that the endpoint IS included in the solution set (used with ≤ or ≥).

Open Parenthesis ( or )

Used in interval notation to indicate that the endpoint is NOT included in the solution set (used with < or >).

Infinity in Interval Notation

Infinity (∞) and negative infinity (−∞) are always written with open parentheses since they are not actual numbers. Example: [1, ∞)

Addition Property of Inequality

If x < y, then x + z < y + z. Like quantities can be added to both sides of an inequality without changing the direction.

Multiplication Property of Inequality — Positive

If x < y and z is positive, then xz < yz. The inequality direction remains the SAME when multiplying by a positive number.

Multiplication Property of Inequality — Negative

If x < y and z is negative, then xz > yz. The inequality direction REVERSES when multiplying or dividing by a negative number.

Compound Inequality

An inequality that requires considering more than one solution interval, connected by the words “and” (intersection) or “or” (union).

Intersection of Sets

The set of all elements common to BOTH sets. Written X ∩ Y. Example: {1,2,3} ∩ {2,3,4} = {2,3}

Union of Sets

The set of all elements that are in EITHER set. Written X ∪ Y. Example: {1,2,3} ∪ {3,4,5} = {1,2,3,4,5}

“And” Compound Inequality

Both conditions must be true simultaneously. The solution is the INTERSECTION of the two individual solution sets.

“Or” Compound Inequality

At least one condition must be true. The solution is the UNION of the two individual solution sets.

Domain of an Expression

The set of all allowable input values for an expression or function; values “in the domain” produce valid outputs.

Outside the Domain

Values that are NOT allowed as inputs. For fractions, any value making the denominator zero is outside the domain.

Domain Restriction — Fractions

Values that cause the denominator to equal zero must be excluded. Example: For 6/(x−2), x ≠ 2, so domain is (−∞, 2) ∪ (2, ∞)

Domain Restriction — Square Roots

The expression under a square root must be ≥ 0. Example: For √(x+3), x + 3 ≥ 0, so x ≥ −3, domain is [−3, ∞)

Absolute Value

The distance of a number from zero on the number line, regardless of direction. Always non-negative. Written

Property of Absolute Value Equations

If

Absolute Value Equation — No Solution

If k < 0, the equation

Absolute Value Equation — One Solution

If k = 0, then

Isolating the Absolute Value

The first step in solving any absolute value equation or inequality — get the absolute value expression alone on one side before applying properties.

Multiplicative Property of Absolute Value

Extraneous Solution

A value obtained algebraically that does NOT satisfy the original equation when substituted back in. Must always be checked and rejected.

Absolute Value Inequality — Less Than

If

Absolute Value Inequality — Greater Than

If

Exponential Function

For b > 0 and b ≠ 1, the function defined by f(x) = b^x. The domain is all real numbers.

Why b > 0 in Exponential Functions

Limiting b to positive values ensures that all outputs f(x) = b^x will be real numbers.

Why b ≠ 1 in Exponential Functions

If b = 1, then y = 1^x = 1 for all x, which is just a constant function — not an exponential function.

Product Rule of Exponents

b^m · b^n = b^(m+n). When multiplying powers with the same base, ADD the exponents.

Quotient Rule of Exponents

b^m / b^n = b^(m−n). When dividing powers with the same base, SUBTRACT the exponents.

Power Rule of Exponents

(b^m)^n = b^(mn). When raising a power to another power, MULTIPLY the exponents.

Product-to-Power Rule

(ab)^n = a^n · b^n. A product raised to a power equals each factor raised to that power.

Negative Exponent Rule

b^(−n) = 1/b^n. A negative exponent means take the reciprocal of the base raised to the positive exponent.

Logarithm Definition

y = log_b(x) means b^y = x. A logarithm is the exponent y to which base b must be raised to produce x.

Fundamental Log Property I

log_b(b) = 1, because b^1 = b. The log of the base itself always equals 1.

Fundamental Log Property II

log_b(1) = 0, because b^0 = 1. The log of 1 is always 0 regardless of the base.

Fundamental Log Property III

log_b(b^x) = x. Applying a logarithm to its own base exponential returns the exponent x directly.

Fundamental Log Property IV

b^(log_b x) = x. Raising a base to its own logarithm returns the original argument x.

Inverse Relationship of Logs and Exponentials

y = log_b(x) and y = b^x are inverse functions of each other. Each operation undoes the other completely.

Product Property of Logarithms

log_b(XY) = log_b(X) + log_b(Y). The log of a product equals the SUM of the individual logarithms.

Quotient Property of Logarithms

log_b(X/Y) = log_b(X) − log_b(Y). The log of a quotient equals the DIFFERENCE of the individual logarithms.

Power Property of Logarithms

log_b(X^p) = p · log_b(X). The log of a power equals the exponent TIMES the logarithm of the base expression.

Change-of-Base Formula

log_b(X) = log(X)/log(b) = ln(X)/ln(b). Converts any logarithm into base 10 or base e for calculator use.

Natural Logarithm

A logarithm with base e ≈ 2.71828. Written ln(x). Used extensively in science and calculus.

Common Logarithm

A logarithm with base 10. Written log(x). The assumed base when no base is written.

Uniqueness Property of Logarithms

If log_b(m) = log_b(n), then m = n. When two logs with the same base are equal, their arguments must be equal.

Solving a Logarithmic Equation

Isolate the log, then exponentiate both sides using the same base to eliminate the logarithm. Always check for extraneous solutions.

Solving an Exponential Equation

Isolate the exponential term, then take the logarithm of both sides to bring the variable out of the exponent.