Algebra Fundamentals and Operations with Integers

Fundamental Concepts

• Algebra is a branch of mathematics that uses letters and symbols to represent numbers and relationships between them.
• Variables are symbols (usually letters) that represent unknown values.
• Constants are fixed numerical values.
• Expressions are combinations of variables, constants, and operations (addition, subtraction, multiplication, division, etc.).
• Equations state that two expressions are equal.
• Inequalities state that two expressions are not equal, using symbols like < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to), or ≠ (not equal to).

Operations with Integers

• Integers are whole numbers (positive, negative, and zero).
• Addition: Combining numbers (e.g., 2 + 3 = 5; -2 + 3 = 1; -2 + (-3) = -5)
• Subtraction: Finding the difference between numbers (e.g., 5 - 2 = 3; 2 - 5 = -3; -2 - 3 = -5; -2 - (-3) = 1)
• Multiplication: Repeated addition (e.g., 3 x 4 = 12; (-3) x 4 = -12; 3 x (-4) = -12; (-3) x (-4) = 12)
• Division: Finding how many times one number is contained in another (e.g., 12 ÷ 3 = 4; -12 ÷ 4 = -3)
• Understanding the rules of signs for multiplication and division(e.g., positive * positive = positive; negative * negative = positive; positive * negative = negative).

Solving Equations

• Equations have an equal sign (=).
• Solving an equation means finding the value of the variable that makes the equation true.
• A common strategy is isolating the variable by using inverse operations (like adding, subtracting, multiplying, or dividing) on both sides of the equation.
• Example: Solve for x in the equation 2x + 5 = 11. Subtract 5 from both sides -> 2x = 6. Divide both sides by 2 -> x = 3, which is the solution.

Simplifying Expressions

• Combining like terms (variables with the same exponent).
• Using the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
• Examples: 3x + 2x = 5x; 2(x+3) = 2x + 6.

Linear Equations and Inequalities in one Variable

• Linear equations have a variable raised to the first power.
• Linear inequalities have the same form as linear equations, but they use inequality symbols.
• Graphing solutions to inequalities: Using number lines and open/closed circles.
  • Open circle for "less than" or "greater than."
  • Closed circle for "less than or equal to" or "greater than or equal to."
• Solving and graphing linear inequalities in one variable (example, find x when 3x + 2 > 8)

Systems of Equations

• A system of equations consists of two or more equations.
• Graphical method involves graphing each equation and finding the point of intersection.
• Substitution method involves solving one equation for a variable and substituting the result into the other equation.
• Elimination method involves adding or subtracting equations to eliminate a variable.
• Solutions can be one point, infinitely many points (if the lines are the same), or no solution (if the lines are parallel).

Polynomials

• Polynomials are algebraic expressions consisting of variables, constants, and exponents.
• Types of polynomials (monomials, binomials, trinomials, etc.)
• Adding, subtracting, multiplying, and dividing polynomials.
• Factoring polynomials (writing a polynomial as a product of simpler polynomials).

Exponents and Radicals

• Understanding exponent rules (product rule, quotient rule, power rule, negative exponents, zero exponents).
• Simplifying expressions with radicals (square roots, cube roots).
• Rationalizing denominators.

Word Problems

• Applying algebraic concepts to solve real-world problems.
• Defining variables, writing equations, and solving for the unknown.
• Common types of word problems (distance, rate, time problems; mixture problems; geometry problems).

This quiz explores basic concepts of algebra, including variables, constants, and expressions. It also covers operations with integers, such as addition, subtraction, and multiplication. Perfect for students looking to strengthen their understanding of algebraic principles.