Algebra Fundamentals Quiz

Fundamental Concepts

• Algebra builds upon arithmetic, using letters (variables) to represent unknown numbers or quantities.
• Variables allow for generalizations of arithmetic rules and problem-solving.
• Equations represent relationships between quantities; solving equations involves finding the value(s) of the variable(s) that satisfy the equation.
• Inequalities compare quantities, showing which is greater or smaller.
• Expressions are combinations of variables, numbers, and operations; simplifying expressions involves combining like terms and following the order of operations (PEMDAS/BODMAS).
• Properties of real numbers (commutative, associative, distributive, identity, inverse) are essential for simplifying expressions and solving equations.

Variables and Expressions

• Variables represent unknown quantities; common variables include x, y, and z.
• Expressions combine variables, numbers, and operations (+, -, ×, ÷).
• Terms are separated by plus or minus signs within an expression.
• Like terms have the same variable(s) raised to the same power; like terms can be combined.
• Constants are terms without variables.
• Simplifying expressions involves applying the order of operations (PEMDAS/BODMAS) and combining like terms.
• Evaluating expressions means substituting values for variables and then calculating the result.

Solving Equations

• Equations state that two expressions are equal.
• Solving an equation means finding the value(s) of the variable(s) that make the equation true.
• The goal is to isolate the variable on one side of the equation through the application of inverse operations on both sides.
• Inverse operations are operations that undo each other (addition and subtraction, multiplication and division).
• Addition property of equality: If a = b, then a + c = b + c.
• Subtraction property of equality: If a = b, then a - c = b - c.
• Multiplication property of equality: If a = b, then a × c = b × c.
• Division property of equality: If a = b, and c ≠ 0, then a ÷ c = b ÷ c.
• Solving multi-step equations often involves several steps using the above properties.

Solving Inequalities

• Inequalities compare quantities using symbols like < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to), or ≠ (not equal to).
• Solving inequalities is similar to solving equations, but the inequality sign is important to consider when multiplying or dividing by a negative number.
• Reversing the direction of the inequality symbol when multiplying or dividing by a negative number is critical.
• Graphing inequalities on a number line helps visualize the solution set.

Linear Equations and Inequalities

• A linear equation has a variable with the highest power of 1; its graph is a straight line.
• Standard form of a linear equation is Ax + By = C.
• Slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
• Linear inequalities have the same form as linear equations, but include inequality symbols.
• Solutions to linear inequalities are often shown as half-planes on a graph.

Systems of Equations

• A system of equations consists of two or more equations with the same variables.
• Solving a system of equations means finding the values of the variables that satisfy all the equations.
• Methods for solving systems of equations include graphing, substitution, and elimination.

Exponents and Polynomials

• Exponents represent repeated multiplication.
• Polynomials are expressions containing variables and exponents, with terms that are single variables or constants.
• The degree of a polynomial is the highest exponent in the expression.
• Polynomial operations (addition, subtraction, multiplication) follow the rules of exponents and combining like terms.

Factoring

• Factoring is the process of expressing a polynomial as a product of simpler factors.
• Factoring is used to solve polynomial equations and simplify expressions.

Radicals

• Radicals represent roots of numbers.
• Understanding properties of radicals (simplified radicals, rationalizing denominators) is important.

Real Numbers

• Real numbers include rational and irrational numbers.
• Rational numbers can be expressed as fractions.
• Irrational numbers cannot be expressed as fractions and have non-repeating, non-terminating decimals.
• Ordering, comparing, and performing operations with real numbers are fundamental to algebra.

Functions

• Functions relate inputs to outputs; functions can be expressed as equations, tables, or graphs.
• Identifying linear and nonlinear functions is a key skill.
• Understanding domain and range is crucial for understanding functions.