Inequalities and Number Properties

Inequality Symbols and Their Meanings

Inequality symbols are used to compare the relative size of numbers or expressions. Understanding these symbols is fundamental in algebra.

• < (less than): Indicates that one value is smaller than another. Example: means "-8 is less than 5."

• ≤ (less than or equal to): Indicates that one value is smaller than or equal to another. Example: means "x is less than or equal to 7."

• > (greater than): Indicates that one value is larger than another. Example: means "x is greater than 4."

• ≥ (greater than or equal to): Indicates that one value is larger than or equal to another. Example: means "x is greater than or equal to -9."

Strict inequalities (<, >) do not include equality, while nonstrict inequalities (≤, ≥) do include equality.

• : x is a positive number

• : x is a negative number

• : x is neutral

• : x is a nonnegative number

• : x is a nonpositive number

Evaluating Algebraic Expressions

Substitution and Absolute Value

To evaluate an algebraic expression, substitute the given values for the variables and perform the indicated operations.

• Absolute value is the distance of a number a from zero on the number line.

Example: If and , evaluate and .

Domain of a Variable

Definition and Determination

The domain of a variable is the set of all possible values that the variable can assume, typically restricted by the context of the problem (such as avoiding division by zero).

• To find the domain, identify values that would make the expression undefined (e.g., denominators equal to zero).

Example: Determine the domain of .

Laws of Exponents

Exponent Rules and Applications

Exponents represent repeated multiplication. The following rules apply for real numbers and integers :

Example: Simplify and .

Additional info: while because of the order of operations.

Polynomials

Definition and Standard Form

A polynomial in one variable is an algebraic expression of the form , where and is a nonnegative integer.

• Leading coefficient: The coefficient of the term with the highest degree.

• Degree: The highest power of the variable in the polynomial.

• Standard form: Terms are written in descending order of degree.

Adding and Subtracting Polynomials

Combine like terms (terms with the same variable and exponent) to add or subtract polynomials.

• Like terms: Terms with the same variable raised to the same power.

Example:

Multiplying Polynomials

Use the distributive property, FOIL method (for binomials), or box method to multiply polynomials. Apply exponent rules as needed.

• FOIL: First, Outer, Inner, Last (for multiplying two binomials).

Example:

Dividing Polynomials

Divide polynomials using long division or synthetic division (when the divisor is linear).

Example: Divide by .

Factoring Polynomials

Factoring Techniques

Factoring is the process of writing a polynomial as a product of its factors. Recognize patterns such as:

• Difference of squares:

• Sum/difference of cubes: ,

• Trinomials: can be factored using the X-method or by inspection.

• Perfect square trinomials:

Example: Factor .

Rational Expressions

Reducing to Lowest Terms

A rational expression is a quotient of two polynomials. Simplify by factoring numerator and denominator and canceling common factors.

Example: Simplify .

Multiplying and Dividing Rational Expressions

• To multiply: Factor, then multiply numerators and denominators, cancel common factors.

• To divide: Multiply by the reciprocal of the divisor.

Example:

Adding and Subtracting Rational Expressions

• Find a common denominator (usually the least common denominator, LCD).

• Rewrite each fraction with the LCD, then add or subtract numerators.

Example:

Simplifying Complex Fractions

• Treat numerator and denominator separately.

• Combine fractions in the numerator and denominator.

• Use the KFC (Keep, Flip, Change) method for division.

Example:

Radicals and Rational Exponents

Simplifying Radicals

The principal nth root of a real number is means .

• Radicand: The number under the radical sign.

• Index: The degree of the root (e.g., 2 for square root, 3 for cube root).

Example: because .

Rationalizing Denominators

To rationalize the denominator, rewrite the expression so that the denominator contains no radicals.

Example:

Simplifying Expressions with Rational Exponents

Recall that .

Example: Simplify .

Summary Table: Laws of Exponents

Key Takeaways

• Mastery of inequalities, exponents, polynomials, and rational expressions is foundational for success in College Algebra.

• Always check for restrictions on the domain, especially when variables appear in denominators or under even roots.

• Factoring and simplifying are recurring skills throughout algebraic problem solving.