R.2 Algebra Essentials

Graphing and Understanding Inequalities

Inequalities are mathematical statements that compare two values or expressions using specific symbols. Understanding these symbols and their meanings is fundamental in algebra.

• Inequality Symbols:

  • < means "is less than" (e.g., means "-8 is less than 5").

  • > means "is greater than" (e.g., means "x is greater than 4").

  • \leq means "is less than or equal to" (e.g., means "x is less than or equal to 7").

  • \geq means "is greater than or equal to" (e.g., means "x is greater than or equal to -9").

• Strict vs. Nonstrict Inequalities:

  • Strict: and (do not include equality)

  • Nonstrict: and (include equality)

• Special Cases:

  • : x is a positive number

  • : x is a negative number

  • : x is a nonnegative number (zero or positive)

  • : x is a nonpositive number (zero or negative)

Absolute Value and Distance on the Real Number Line

The absolute value of a number is its distance from zero on the real number line, regardless of direction. It is always nonnegative.

• Definition: is the distance from 0 to on the number line.

• Examples:

Evaluating Expressions with Absolute Value

To evaluate expressions involving absolute value, substitute the given values and apply the definition of absolute value.

• Example 1: Evaluate each expression if and .

  • a)

  • b)

• Solution Steps:

  1. Substitute the values for and .

  2. Compute inside the absolute value first.

  3. Apply the absolute value, then complete the arithmetic operations.

Domain of a Variable

The domain of a variable is the set of all possible values that the variable can take so that the expression is defined (i.e., does not involve division by zero or taking the square root of a negative number in real numbers).

• Division by Zero: Expressions with denominators cannot have values that make the denominator zero.

• Example: Typing in a calculator is undefined.

Examples: Determining the Domain

• Example 2: Determine the domain of the variable in each expression. Use set notation.

  • a) Domain: All real numbers except (since denominator cannot be zero). Set notation:

  • b) Domain: All real numbers except and . Set notation:

  • c) Domain: All real numbers (since is never zero for real ).

Laws of Exponents

Exponents are used to represent repeated multiplication. The following laws help simplify expressions involving exponents.

• Product Rule:

• Quotient Rule:

• Power Rule:

• Power of a Product:

• Power of a Quotient:

• Zero Exponent: (for )

• Negative Exponent:

• Negative Exponents in Fractions:

• Negative Power of a Quotient:

Examples:

• (for )

Table: Laws of Exponents

Simplifying Expressions with Exponents

When simplifying expressions, always express the final answer with positive exponents. Assume the base is not zero when dealing with zero or negative exponents.

• Example 3: Simplify each expression. Express the answer so that all exponents are positive.

  • a) Solution:

  • b) Solution:

  • c) Solution: Apply the power to each factor:

    • ,

    • ,

    • ,

    • ,

    Combined:

  • d) Solution:

    • Numerator:

    • Denominator:

    • Combine:

    • Simplify:

    • With positive exponents:

  • e) Solution:

    • First, simplify inside the parentheses:

    • So,

    • Now, raise to the power:

    • Numerator: to the power becomes

    • Denominator:

    • So,

    • Final answer:

Additional info: When simplifying expressions with exponents, always combine like bases and apply exponent rules systematically. Remember that a negative exponent indicates the reciprocal, and zero exponents (except for ) always yield 1.