Real Number Properties and Linear Inequalities

Real Number Properties

• Zero is the additive identity: Adding zero to any number results in the number itself. For example, 0 + 2 = 2.
• One is the multiplicative identity: Multiplying any number by one results in the number itself. For example, 1 * 2 = 2.
• Each real number has an additive inverse: Given a real number x, its additive inverse is –x. Adding a number to its additive inverse always results in zero. For example, 4 + (-4) = 0.
• Each real number has a multiplicative inverse, except for zero: Given any real number x that is not zero, its multiplicative inverse is 1/x. Multiplying any number by its multiplicative inverse always results in 1. For example, 4 * 1/4 = 1.
• The distributive property of multiplication over addition: Multiplying a number by a sum is the same as multiplying the number by each term in the sum and adding the products together. For example, 2(3+4) = 23 + 24.

Linear inequalities

• A mathematical statement indicating the relative order of two expressions using symbols like <, >, ≤, or ≥.
• Can be solved by performing the same operations on both sides of the inequality, keeping in mind that multiplying or dividing by a negative number flips the direction of the inequality sign.

Functions

• Definition: A function is a rule that assigns a unique output value to each input value. It can be expressed as a set of ordered pairs, an equation, or a graph.
• Finding the domain of a function: The domain of a function is the set of all possible input values for which the function is defined. For example, the domain of a function with a square root can be determined by ensuring the expression under the square root is non-negative.

Evaluating Functions

• To evaluate a function, substitute the given input value into the function's rule and simplify the resulting expression.
• For example, if f(x) = x^2 + 1, then f(2) = 2^2 + 1 = 5.
• Understand what each part of the function represents:
  • An independent variable is the input value (often denoted by x).
  • A dependent variable is the output value (often denoted by y or f(x)).

Exponents and Radicals

• Integer exponents: A positive integer exponent indicates repeated multiplication of the base. A negative integer exponent indicates taking the reciprocal of the base raised to the positive version of the exponent. For example, 2^5 = 22222 = 32, and 2^-3 = 1/2^3 = 1/8.
• Exponent Properties:
  • Product of Powers : 𝑥^𝑚 * 𝑥^𝑛 = 𝑥^(𝑚+𝑛)
  • Power of a Power : (𝑥^𝑚)^𝑛 = 𝑥^(𝑚*𝑛)
  • Power of a Product : (𝑥*𝑦)^𝑛 = 𝑥^𝑛 * 𝑦^𝑛
• Simplifying expressions:
  • Combine exponents with the same base using the exponent properties.
  • Express final answers using positive exponents.
• Radicals: A radical symbol (√) represents a root of a number. The index of a radical gives the order of the root.
  • Square Root: (√x) represents the number that, when multiplied by itself, gives x. For example, √9 = 3.
  • Cube Root: (³√x) represents the number that, when multiplied by itself three times, gives x. For example, ³√8 = 2.
• Simplifying expressions:
  • Find the prime factorization of the number under the radical.
    • If there are multiple factors with the same exponent, take out one factor to the outside of the radical and reduce the exponent under the radical.
    • If the exponent of a factor divides the index of the radical, take out the entire factor.

This quiz covers fundamental properties of real numbers, including additive and multiplicative identities, inverses, and the distributive property. Additionally, it explores the concept of linear inequalities and their implications in mathematics.